Radicals of Morita context rings

Abstract

For a Morita context ring T=(\genfrac{}{}{0ex}{}{RN}{MS}), the structure of several radicals is given under certain conditions.

Keywords

Morita context ring / radical

1 Introduction

Let R,S be two rings, {}_R{N}_S,{}_S{M}_Rbe two bimodules, and (\cdot ,\cdot ):N{\otimes }_{S}M\to R and [\cdot ,\cdot ]:M{\otimes }_{R}N\to S be two bimodule homomorphisms satisfying the conditions of

(n,m)n^{\prime }=n[m,n^{\prime }],[m,n]m^{\prime }=m\left(n,m^{\prime }\right),\mathrm{\forall }m,m^{\prime }\in M,\mathrm{\forall }n,n^{\prime }\in N,

then the system (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) is called a Morita context. Let

T=\left(\begin{array}{cc}R& N\\ M& S\end{array}\right)=\left\{\left(\begin{array}{cc}r& n\\ m& s\end{array}\right)|r\in R,s\in S,n\in N,m\in M\right\}.

With respect to matrix addition, T is an Abel group. Define multiplication by

\left(\begin{array}{cc}r& n\\ m& s\end{array}\right)\left(\begin{array}{cc}{r}^{\prime }& n^{\prime }\\ m^{\prime }& s^{\prime }\end{array}\right)=\left(\begin{array}{cc}r{r}^{\prime }+\left(n,m^{\prime }\right)& r{n}^{\prime }+n{s}^{\prime }\\ m{r}^{\prime }+s{m}^{\prime }& \left[m,n^{\prime }\right]+s{s}^{\prime }\end{array}\right),

and then T becomes an associative ring, called a Morita context ring. The images of the two bimodule homomorphisms (N,M)=NM and [M,N]=MN refer to the ideals of R,S, respectively, called the trace ideals of this Morita context. If A,B,V,W, are subsets of R,S,N,M, respectively, then \left(\begin{array}{cc}A& V\\ W& B\end{array}\right) refers to the subset \left\{\left(\begin{array}{cc}a& v\\ w& b\end{array}\right)|a\in A,b\in B,v\in V,w\in W\right\} of \left(\begin{array}{cc}R& N\\ M& S\end{array}\right). When there is 1 in R,S and N,M are unital modules, if I=\left(\begin{array}{cc}A& V\\ W& B\end{array}\right) is an ideal of T, then A,B are ideals of R,S, respectively, and V,W, are sub-bimodules of N,M respectively.

Generally, let R_1,R_2,\dots ,R_n be n rings (n\ge 2), and M_{ij} be R_i\text{-}{R}_j\text{-} bimodules such that M_{ii}=R_i,i,j=1,2,\dots ,n. For any i,j,k=1,2,\dots ,n,i\ne j,k\ne j, there exist R_i\text{-}{R}_j\text{-}bimodule homomorphisms

{\phi }_{ikj}:M_{ik}{\otimes }_{R_k}{M}_{kj}\to M_{ij}.

For i=j,k=j, there are canonical isomorphisms

{\phi }_{iij}:R_i{\otimes }_{R_i}{M}_{ij}\to M_{ij},

{\phi }_{ijj}:M_{ij}{\otimes }_{R_j}{R}_j\to M_{ij}.

Define ab={\phi }_{ikj}(a\otimes b), satisfying the associative law (ab)c=a(bc) for all a\in M_{ik},b\in M_{kj},c\in M_{jl} and all i,j,k,l. Let T be the set of all n\times n matrices \left\{\left(a_{ij}\right)|a_{ij}\in M_{ij},\mathrm{\forall }1\le i,j\le n\right\}. With respect to the addition and multiplication of matrices, T becomes an associative ring, called a n\times n Morita context ring.

For the n\times n Morita context ring, let

R=R_1,

N=\left(M_{12},M_{13},\dots ,M_{1n}\right),

M=\left(\begin{array}{c}{M}_{21}\\ M_{31}\\ ⋮\\ M_{n1}\end{array}\right),

S=\left(\begin{array}{cccc}{R}_2& M_{23}& \dots & M_{2n}\\ M_{32}& R_3& \dots & M_{3n}\\ ⋮& ⋮& \ddots & ⋮\\ M_{n2}& M_{n3}& \dots & R_n\\ & & & \end{array}\right).

Then S itself is a (n-1)\times (n-1) Morita context ring. On the row and column multiplication of matrices, N is an R\text{-}S\text{-}bimodule and M indicates an S\text{-}R\text{-}bimodule. According to the multiplication that {\phi }_{ikj} defines on T, there are bimodule homomorphisms N{\otimes }_{S}M\to R and M{\otimes }_{R}N\to S, satisfying the associative law. Thus, T\cong (\genfrac{}{}{0ex}{}{RN}{MS}). Next, it focuses on the discussion of 2×2 Morita context rings.

Various matrix rings play important roles in ring theory. Morita context rings generalize the concept of matrix rings over a given ring. As a main tool in the theory of module category equivalence, it has been widely used in algebraic branches. Every ring A with a non-trivial idempotent element e is isomorphic to the Morita context ring determined by the Morita context Morita context (eAe,(1-e)A(1-e),(1-e)Ae,eA(1-e)). The following shows examples of Morita contexts.

(1) Let R be a ring, {}_{R}M be a left R-module, and M^{\ast }=\text{Hom}(M,R),E={\text{End}}_{R}M, then \left(R,E,M^{\ast },M\right) forms a Morita context.

(2) Let G be a finite group acting on R, R^G=\left\{x\in R\mid x^g=x,\mathrm{\forall }g\in G\right\} be the stable ring, and R\ast G=\left\{\begin{array}{r}{\sum }_{g\in G}{r}_{g}g\mid r_g\in R\end{array}\right\} be the skew group ring. Among them, the multiplication is rg\cdot sh=r{s}^{g^{-1}}gh,\mathrm{\forall }g,h\in G,r,s\in R. Define the multiplication as

x\cdot r={\sum }_{g\in G}{r}_g{r}^{g^{-1}},r\cdot x={\sum }_{g\in G}{\left(r{r}_g\right)}^g,\mathrm{\forall }x={\sum }_{g\in G}{r}_{g}g,\mathrm{\forall }r\in R,

then R becomes (R^G,R^G)- bimodule as well as (R\ast G,R\ast G)- bimodule. (R^G,R\ast G,R,R) forms a Morita context.

(3) Let H be a finite-dimensional Hopf algebra, and there exists a S-fixed integral {\int }_l,0\ne t\in {\int }_l,\lambda \in H^{\ast } such that th=\lambda (h)t, \mathrm{\forall }h\in H. For a\mathrm{♯}h\in A\mathrm{♯}H,b\in A, define

(a\mathrm{♯}h)\cdot b=a(h\cdot b),b\cdot (a\mathrm{♯}h)=\overline{S}{h}^{\lambda }\cdot (ab),

where \overline{S} is the composition inverse of the antipode S. Then (A^H,A\mathrm{♯}H,{}_{A\mathrm{♯}H}{A}_{A^H},{}_{A^H}{A}_{A\mathrm{♯}H}), with respect to the maps [\cdot ,\cdot]: A{\otimes }_{A^H}A\to A\mathrm{♯}H, [a,b]=atb and (\cdot ,\cdot ):A{\otimes }_{A\mathrm{♯}H}A\to A^H,(a,b)=t\cdot (ab), form a Morita context.

(4) Let (\mathrm{\Gamma },M) be a weak Nobusawa \mathrm{\Gamma }-ring. \mathrm{\Gamma }M represents \mathrm{\Gamma }{\otimes }_{\text{Z}}M, and M\mathrm{\Gamma } is \text{M}{\otimes }_{\text{Z}}\mathrm{\Gamma }. Then

\left(\begin{array}{cc}\mathrm{\Gamma }M& \mathrm{\Gamma }\\ M& M\mathrm{\Gamma }\end{array}\right)

is a Morita context ring. Conversely, for any Morita pair (Q,R,{}_R{T}_Q,{}_Q{S}_R), there is Morita context \mathrm{\Gamma }- ring (S,T).

There are many studies on Morita context rings. For example, Amitsur [1] discussed the relationship between the properties of T and those of R,S,N,M. Poole and Stewart [14] gave the canonical quotient ring of Morita context rings. Chen et al. [4] characterized the ideal lattice of Morita context rings. In particular, Jaegermann [11] explored the properties of the strong radical and hereditary radical of rings via Morita context. Nicholson and Watters [13] studied normal radicals and normal classes based on Morita context. In this paper, we promote the research in [11] and [13], and provide specific constructions of radicals for Morita context rings.

It is assumed that there is 1 for all rings discussed in this paper, and all modules are unital modules. Let I be an ideal of ring A. Iis called a prime ideal of A if for any two ideals B,C of A, and BC\subseteq I implies that BC\subseteq I or C\subseteq I. A ring whose zero ideal is a prime ideal is called a prime ring. The intersection of all prime ideals of A, denoted by P(A), is called the prime radical of A. Iindicates a completely prime ideal of A. If for any two elements a,b of A, ab\in I implies that a\in I or b\in I. According to [17], the intersection of all completely prime ideals of A, denoted by P_2(A), is called the generalized nilradical of A. The Jacobson radical J(A) of ring A is the intersection of all maximal left ideals of A. The simple radical S(A) of A is defined as the intersection of all maximal ideals of A. The Brown-McCoy radical g(A) of A is the intersection of all maximal ideals I_{\alpha } of A such that A/I_{\alpha } is a simple ring. For other concepts of radicals not explained in this paper, refer to [17].

2 Generalized nilradical, strongly prime radical, and Brown-McCoy radical

Handelman and Lawrence [9] defined the concept of strongly prime rings. A finite subset F of A is called a right insulator of A. If for any r\in A, Fr=0 implies r=0. A ring A is called right strongly prime if every nonzero ideal of A contains a right insulator. The right strongly prime radical of A, denoted by s_r(A), is defined as the intersection of all ideals of A such that A/I is right strongly prime, then such an ideal I is called a right strongly prime ideal of A. The left strongly prime radical (denoted by s_l(A)) and left strongly prime ideals of Aare defined similarly.

Proposition 1.1　 If the trace ideals NM and MN of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) are both nilpotent, then

(1) assume

I=\left(\begin{array}{cc}A& U\\ V& B\end{array}\right)

is the prime ideal of

T=\left(\begin{array}{cc}R& N\\ M& S\end{array}\right),

then there must be U=N,V=M, and A\supseteq NM,B\supseteq MN are prime ideals of R,S, respectively;

(2) let I be a completely prime ideal of T. Then there must be U=N,V=M, and A\supseteq NM,B\supseteq MN are completely prime ideals of R,S, respectively;

(3) let I be a right strongly prime ideal of T. Then there must be U=N,V=M, and A\supseteq NM,B\supseteq MN are right strongly prime ideals of R,S, respectively.

Proof　(1) Let I=\left(\begin{array}{cc}A& U\\ V& B\end{array}\right) be a prime ideal of T. Take J=\left(\begin{array}{cc}NM& N\\ M& MN\end{array}\right). It is verified that J is an ideal of T, and there are

\begin{array}{rl}& J^3\subseteq \left(\begin{array}{cc}(NM{)}^3+(NM{)}^2& (NM{)}^{2}N+(NM)N\\ (MN{)}^{2}M+(MN)M& (MN{)}^3+(MN{)}^2\end{array}\right),\\ & J^5\subseteq \left(\begin{array}{cc}(NM{)}^5+(NM{)}^4+(NM{)}^3& (NM{)}^{4}N+(NM{)}^{3}N+(NM{)}^{2}N\\ (MN{)}^{4}M+(MN{)}^{3}M+(MN{)}^{2}M& (MN{)}^5+(MN{)}^4+(MN{)}^3\end{array}\right),\\ & J^7\subseteq \left(\begin{array}{cc}(NM{)}^7+(NM{)}^6+\cdots +(NM{)}^4& (NM{)}^{6}N+(NM{)}^{5}N+\cdots +(NM{)}^{3}N\\ (MN{)}^{6}M+(MN{)}^{5}M+\cdots +(MN{)}^{3}M& (MN{)}^7+(MN{)}^6+\cdots +(MN{)}^4\end{array}\right),\\ & ⋮\\ & J^{2n-1}\subseteq \left(\begin{array}{cc}(NM{)}^{2n-1}+\cdots +(NM{)}^n& (NM{)}^{2n-2}N+\cdots +(NM{)}^{n-1}N\\ (MN{)}^{2n-2}M+\cdots +(MN{)}^{n-1}M& (MN{)}^{2n-1}+\cdots +(MN{)}^n\end{array}\right),n\in \mathbb{N}.\end{array}

Since NM and MN are nilpotent, there exists m\in N such that (NM{)}^m=0,(MN{)}^m=0. It implies J^{2m-1}\subseteq \left(\begin{array}{cc}0& N_1\\ M_1& 0\end{array}\right), where M_1\subseteq M,N_1\subseteq N. Then {\left(J^{2m-1}\right)}^{2m}=0, so J is a nilpotent ideal of T. Therefore J\subseteq I, and hence I=\left(\begin{array}{cc}{A}_1& N\\ M& B_1\end{array}\right), where A_1\supseteq NM,B_1\supseteq MN. From IT\subseteq I and TI\subseteq I, it follows that A_1⊲A,B_1⊲B.

It is assumed that A_2◃R,A_3◃R, and A_2\supseteq NM,A_3\supseteq MN such that A_2{A}_3\subseteq A_1. Then I_2=\left(\begin{array}{cc}{A}_2& N\\ M& B_1\end{array}\right) and I_3=\left(\begin{array}{cc}{A}_3& N\\ M& B_1\end{array}\right) are ideals of T, and there is

I_2{I}_3\subseteq \left(\begin{array}{cc}{A}_2{A}_3+NM& N\\ M& MN+B\end{array}\right)\subseteq \left(\begin{array}{cc}{A}_1& N\\ M& B_1\end{array}\right)=I.

Since I is prime, it follows that I_2\subseteq I or I_3\subseteq I, hence A_2\subseteq A_1 or A_3\subseteq A_1. Therefore A_1 is a prime ideal of R. Similarly, B_1 is a prime ideal of S.

(2) Let I be any completely prime ideal of T. Then I is a prime ideal of T. By (1), I=\left(\begin{array}{cc}A& N\\ M& B\end{array}\right), where A\supseteq NM,B\supseteq MN are prime ideals of R,S, respectively.

For a_1,a_2\in A,b_1,b_2\in B, if a_1{a}_2\in A,B_1{B}_2\in B, then for any v_1{v}_2\in N,w_1{w}_2\in M, there is

\left(\begin{array}{cc}{a}_1& v_1\\ w_1& b_1\end{array}\right)\left(\begin{array}{cc}{a}_2& v_2\\ w_2& b_2\end{array}\right)=\left(\begin{array}{cc}{a}_1{a}_2+v_1{w}_2& v_3\\ w_3& w_1{v}_2+b_2\end{array}\right)\in \left(\begin{array}{cc}A& N\\ M& B\end{array}\right)=I.

Since I is completely prime, it follows that \left(\begin{array}{cc}{a}_1& v_1\\ w_1& b_1\end{array}\right)\in I or \left(\begin{array}{cc}{a}_2& v_2\\ w_2& b_2\end{array}\right)\in I, hence a_1\in A,b_1\in B or a_2\in A,b_2\in B, which shows that A,B are completely prime ideals of R,S, respectively.

(3) Let I_1 be any right strongly prime ideal of T, so I_1 is a prime ideal of T. By (1), I_1=\left(\begin{array}{cc}{A}_1& N\\ M& B_1\end{array}\right), where A_1\supseteq NM,B_1\supseteq MN are prime ideals of R,S, respectively. The following shows that A_1,B_1 are also right strongly prime ideals.

R/A_1 is a prime ring. Take any nonzero ideal A_2/A_1 of R/A_1. Then

A_2⫌A_1,I_2=\left(\begin{array}{cc}{A}_2& N\\ M& B_1\end{array}\right)⫌\left(\begin{array}{cc}{A}_1& N\\ M& B_1\end{array}\right).

Since T/I_1 is right strongly prime and 0\ne I_2/I_1◃T/I_1,I_2/I_1\cong A_2/A_1 contain finite insulators

F=\left(\begin{array}{cc}{A}_0/A_1& 0\\ 0& 0\end{array}\right)\subseteq \left(\begin{array}{cc}{A}_2/A_1& 0\\ 0& 0\end{array}\right),

where A_0/A_1 is a finite set, such that for any \left(\begin{array}{cc}\overline{a}& 0\\ 0& 0\end{array}\right)\in T/I_1\cong A_0/A_1\oplus B_0/B_1, if

\left(\begin{array}{cc}{A}_0/A_1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{cc}\overline{a}& 0\\ 0& 0\end{array}\right)=0,

then there must be \overline{a}=0. Equivalently, if \left(A_0/A_1\right)\overline{a}=0, then \overline{a}=0. It shows that A_0/A_1\subseteq A_2/A_1 is a finite insulator for A_2/A_1, so R/A_1 is right strongly prime, and A_1 is a right strongly prime ideal. Similarly, B_1 is a right strongly prime ideal.

Proposition 1.2　(1) Let A_{\alpha },B_{\beta } be prime ideals of R,S, respectively. Then

I_{\alpha }=\left(\begin{array}{cc}{A}_{\alpha }& N\\ M& S\end{array}\right),J_{\beta }=\left(\begin{array}{cc}R& N\\ M& B_{\beta }\end{array}\right)

are prime ideals of T;

(2) Let A_{\alpha },B_{\beta } be completely prime ideals of R,S, respectively. Then I_{\alpha },J_{\beta } are completely prime ideals of T;

(3) Let A_{\alpha },B_{\beta } be right strongly prime ideals of R,S, respectively. Then I_{\alpha },J_{\beta } are right strongly prime ideals of T.

Proof　(1) Let I=\left(\begin{array}{cc}{A}_1& V_1\\ W_1& B_1\end{array}\right),J=\left(\begin{array}{cc}{A}_2& V_2\\ W_2& B_2\end{array}\right) be two ideals of T such that IJ\subseteq I_{\alpha }. Then A_1{A}_2+V_1{W}_2\subseteq A_{\alpha }. Hence A_1{A}_2\subseteq A_{\alpha }. Since A_1,A_2are ideals of R, it follows that A_1\subseteq A_{\alpha } or A_2\subseteq A_{\alpha }. Thus, I\subseteq I_{\alpha } or J\subseteq I_{\alpha }. As a result, I_{\alpha } is a prime ideal of T. Similarly, J_{\beta } is a prime ideal of T.

(2) The proof is similar to (1).

(3) By (1), I_{\alpha }=\left(\begin{array}{cc}{A}_{\alpha }& N\\ M& B\end{array}\right),J_{\beta }=\left(\begin{array}{cc}A& N\\ M& B_{\beta }\end{array}\right) are prime ideals of T. Take any nonzero ideal I_1/I_{\alpha }=\left(\begin{array}{cc}{A}_1/A_{\alpha }& 0\\ 0& 0\end{array}\right) of T/I_{\alpha }\cong A/A_{\alpha }. Then A_1⫌A_{\alpha },A_1/A_{\alpha } is a nonzero ideal of A/A_{\alpha }. Since A_{\alpha } is a right strongly prime ideal of A, there exists a finite subset A_0/A_{\alpha }\subseteq A_1/A_{\alpha } such that for any \overline{a}\in A_0/A_{\alpha }, if \left(A_0/A_{\alpha }\right)\overline{a}=0, there must be \overline{a}=0. For any \left(\begin{array}{cc}\overline{a}& 0\\ 0& 0\end{array}\right)\in T/I_{\alpha }\cong A/A_{\alpha }, if

\left(\begin{array}{cc}{A}_0/A_{\alpha }& 0\\ 0& 0\end{array}\right)\left(\begin{array}{cc}\overline{a}& 0\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}\left(A_0/A_{\alpha }\right)\overline{a}& 0\\ 0& 0\end{array}\right)=0.

There must be \overline{a}=0. Therefore, \left(\begin{array}{cc}\left(A_0/A_{\alpha }\right)\overline{a}& 0\\ 0& 0\end{array}\right)=0 is a finite insulator for I_1/I_{\alpha }, and I_{\alpha } is a right strongly prime ideal of T. Similarly, J_{\beta } is a right strongly prime ideal of T.

Theorem 1.1　 If the trace ideals NM and MN of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) are both nilpotent, then

(1) the generalized nilradical of T is

P_2(T)=\left(\begin{array}{cc}{P}_2(R)& N\\ M& P_2(S)\end{array}\right);

(2) the right strongly prime radical of T is

s_r(T)=\left(\begin{array}{cc}{s}_r(R)& N\\ M& s_r(S)\end{array}\right).

Proof　(1) Let Q=\left(\begin{array}{cc}{P}_2(R)& N\\ M& P_2(S)\end{array}\right). Since T/Q\cong R/P_2(R)\oplus S/P_2(S), T/Q is a P_{2^{-}} semisimple ring, then Q\supseteq P_2(T).

On the other hand, by Proposition 1.1 (2), there is

\begin{array}{rl}{P}_2(T)& =\bigcap _{\alpha }{P}_{\alpha }\left(P_{\alpha }\text{runs over all completely prime ideals of}T\right)\\ & \supseteq \bigcap _{\alpha }\left(\begin{array}{cc}{A}_{\alpha }& N\\ M& B_{\alpha }\end{array}\right)\left(A_{\alpha }\supseteq NM,B_{\alpha }\supseteq MN\text{are completely prime ideals of}A,B\right)\\ & =\left(\begin{array}{cc}{\bigcap }_{\alpha }{A}_{\alpha }& N\\ M& {\bigcap }_{\alpha }{B}_{\alpha }\end{array}\right)\supseteq \left(\begin{array}{cc}{P}_2(R)& N\\ M& P_2(S)\end{array}\right)=Q,\end{array}

so P_2(T)=Q.

(2) The proof is similar to (1).

Proposition 1.3　 It is assumed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq J(R),MN\subseteq J(S). Then

(1)I=\left(\begin{array}{cc}A& U\\ V& B\end{array}\right)

is a maximal left ideal of T if and only if

I=\left(\begin{array}{cc}{A}_1& N\\ M& S\end{array}\right)orI=\left(\begin{array}{cc}R& N\\ M& B_1\end{array}\right),

where A_1\supseteq NM,B_1\supseteq MN are maximal left ideals of R,S,respectively.

(2) Iis a maximal ideal of T if and only if I=\left(\begin{array}{cc}{A}_1& N\\ M& S\end{array}\right) or I=\left(\begin{array}{cc}R& N\\ M& B_1\end{array}\right), where A_1\supseteq NM,B_1\supseteq MN are maximal ideals of R,S,respectively.

Proof　(1) Necessity: Let I=\left(\begin{array}{cc}{A}_1& Y\\ X& B_1\end{array}\right) be a maximal left ideal of T=\left(\begin{array}{cc}R& N\\ M& S\end{array}\right). Then there is

R{A}_1+NX\subseteq A_1,RY+N{B}_1\subseteq Y,M{A}_1+SX\subseteq X,MY+S{B}_1\subseteq B_1,

which implies that A_1,B_1 are left ideals of R,S,respectively. Consider two cases:

(i) If A_1\ne R, then since R has an identity, there exists a maximal left ideal A_2\supseteq A_1 of R. According to the assumption

NM\subseteq J(R)=\bigcap _{\alpha }{J}_{\alpha }\left(J_{\alpha }\text{runs over all maximal left ideals of}R\right)

there is NM\subseteq A_2, and there is a left ideal \left(\begin{array}{cc}{A}_2& N\\ M& S\end{array}\right)\supseteq \left(\begin{array}{cc}{A}_1& Y\\ X& B_1\end{array}\right) of T. According to the maximality of I, there must be

A_2=A_1,Y=N,X=M,B_1=S,

namely I=\left(\begin{array}{cc}{A}_2& N\\ M& S\end{array}\right).

(ii) If A_1=R, then according to M{A}_1+SX\subseteq X, there is MR\subseteq X. Since there is 1 for Rand M\subseteq X, then X=M. Moreover, it is known from MN\subseteq J(S) that \left(\begin{array}{cc}R& N\\ M& B_1\end{array}\right)\left(\supseteq \left(\begin{array}{cc}R& Y\\ M& B_1\end{array}\right)\right) is a left ideal of T. According to the maximality of I, there is Y=N and I=\left(\begin{array}{cc}R& N\\ M& B_1\end{array}\right). Let B_2 be a maximal left ideal of S and B_2\supseteq B_1. Then B_2\supseteq J(S)\supseteq MN. It is verified that J=\left(\begin{array}{cc}R& N\\ M& B_2\end{array}\right)\supseteq I is a left ideal of T. By the maximality of I, there is B_2=B_1.

The sufficiency is straightforward.

(2) The proof is similar to (1).

Theorem 1.2　 It is assumed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq J(R),MN\subseteq J(S). Then

(1) the simple radical of T is

S(T)=\left(\begin{array}{cc}S(R)& N\\ M& S(S)\end{array}\right);

(2) the Brown-McCoy radical of T is

g(T)=\left(\begin{array}{cc}g(R)& N\\ M& g(S)\end{array}\right).

Proof　It can be obtained from the definitions of the simple radical and the Brown-McCoy radical and Proposition 1.3.

Corollary 1.1　 If the trace ideals NMand MN of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) are both nilpotent, then

(1) T is a semiprime ring if and only if R,S are semiprime rings and N=0,M=0;

(2) if T is a prime ring, then R,S are prime rings and N=0,M=0.

Proof　(1) T is a semiprime ring if and only if P(T)=0. By Theorem 1.1 (1), it holds if and only if N=0,M=0 and R,S are semiprime rings.

(2) If T is a prime ring, then the zero ideal of T is prime. By Proposition 1.1 (1), it follows that N=0,M=0 and R,S are prime rings.

The converse of Corollary 1.1 (2) is not true. For example,

I=\left(\begin{array}{cc}0& 0\\ 0& S\end{array}\right),J=\left(\begin{array}{cc}R& 0\\ 0& 0\end{array}\right)

are both ideals of T=\left(\begin{array}{cc}R& 0\\ 0& S\end{array}\right) and IJ=0, but I,J are nonzero. Thus, the zero ideal is not prime in T.

A ring R is called a left quasi-pseudo ring [15] if every maximal left ideal of R is an ideal of R.

Corollary 1.2　 Suppose the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq J(R),MN\subseteq J(S). Then T is a left quasi-pseudo ring if and only if R,S are left quasi-pseudo rings.

Proof　It is assumed that R,S are left quasi-pseudo rings. According to Proposition 1.3 (1), every maximal left ideal of T is an ideal of T, so T is a left quasi-pseudo ring. Conversely, it is supposed thatTis a left quasi-pseudo ring. Then for any maximal left ideals A of R and B of S, R+NM,S+MN are left ideals of R,S respectively. Based on the maximality of A,B, there is NM\subseteq A,MN\subseteq B. Then by Proposition 1.3 (1), I=\left(\begin{array}{cc}A& N\\ M& S\end{array}\right) and J=\left(\begin{array}{cc}R& N\\ M& B\end{array}\right) are maximal left ideals of T=\left(\begin{array}{cc}R& N\\ M& S\end{array}\right), hence they are ideals of T, and A,B are ideals of R,S respectively. R,S are left quasi-pseudo rings.

A ring R is called periodic [8], if for every element x of R, there exist two distinct positive integers m,n such that x^m=x^n. A ring R is called N-nil [16], if for any x\in R, there exist positive integers n,k depending on x such that (nx{)}^k=0. A ring R is called a p-ring [16] if for any a\in R, there exists a polynomial f(x)=k_n{x}^n+k_{n-1}{x}^{n-1}+\cdots +k_{1}x with zero constant term such that f(a)=0. Periodicity, N-nil property and the p-ring property are both radical properties. The periodic radical P_r(R) of ring R is the sum of all periodic ideals of R. The N-nil radical K_N(R) of ring R is the sum of all N-nil ideals of R. The p-radical p_f(R) of ring R is the sum of all p-ideals of R.

Theorem 1.3　 It is assumed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM=0,MN=0. Then

(1) the nil radical of T is

K(T)=\left(\begin{array}{cc}K(R)& N\\ M& K(S)\end{array}\right);

(2) the p-radical of T is

p_f(T)=\left(\begin{array}{cc}{p}_f(R)& N\\ M& p_f(S)\end{array}\right);

(3) the N-nil radical of T is

K_N(T)=\left(\begin{array}{cc}{K}_N(R)& N\\ M& K_N(S)\end{array}\right);

(4) the periodic radical of T is

P_r(T)=\left(\begin{array}{cc}{P}_r(R)& N\\ M& P_r(S)\end{array}\right).

Proof　(1) According to [16, Theorem 2.1], T is a nil ring if and only if R,S are nil rings. Thus, as the largest nil ideal of T, K(T)=\left(\begin{array}{cc}K(R)& N\\ M& K(S)\end{array}\right).

(2)‒(3) can be obtained from [16, Theorems 2.4‒2.5].

(4) The proof is similar to (1).

3 Generalized Prime Radical, Singular Radical, and Behrens Radical

A ring R is called a \text{2}-prime ring [3], if the prime radical P(R) of R equals the set \text{Nil}(R) of all nilpotent elements of R, namely P(R)=\mathrm{N}\mathrm{i}\mathrm{l}(R). In [3], Birkenmeier et al. systematically studied 2-prime rings and proved that one of its sub-classes {\mathrm{\Re }}^2=R every prime ideal of R is completely prime\} forms an Amitsur-Kurosh root, called the generalized prime radical. The generalized prime radical {\mathfrak{P}}_c(R) of a ring R is the sum of all ideals I such that I\in {\mathrm{\Re }}^2.

Theorem 2.1　 It is assumed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM=0,MN=0. Then the generalized prime radical of T is

{\mathfrak{P}}_c(T)=\left(\begin{array}{cc}{\mathfrak{P}}_c(R)& N\\ M& {\mathfrak{P}}_c(S)\end{array}\right).

Proof　Let \mathfrak{P}=\left(\begin{array}{cc}{\mathfrak{P}}_c(R)& N\\ M& {\mathfrak{P}}_c(S)\end{array}\right). Since T/\mathfrak{P}\cong R/{\mathfrak{P}}_c(R)\oplus S/{\mathfrak{P}}_c(S) andT/\mathfrak{P} is {\mathfrak{P}}_c- semisimple, then \mathfrak{P}\supseteq {\mathfrak{P}}_c(T). On the other hand, by Proposition 1.1 (1), for the ideal P_1=\left(\begin{array}{cc}{\mathfrak{P}}_c(R)& N\\ M& 0\end{array}\right) of T, every prime ideal of {\mathfrak{P}}_1 has the form I=\left(\begin{array}{cc}{A}_1& N\\ M& 0\end{array}\right), where A_1 is a prime ideal of {\mathfrak{P}}_c(R). However, the ring {\mathfrak{P}}_c(R)\in R^2, so A_1 is a completely prime ideal and I is a completely prime ideal of {\mathfrak{P}}_1. It proves that {\mathfrak{P}}_1 is a R^2- ring and {\mathfrak{P}}_1\subseteq {\mathfrak{P}}_c(T). Similarly, {\mathfrak{P}}_2=\left(\begin{array}{cc}0& N\\ M& {\mathfrak{P}}_c(S)\end{array}\right) is also a R^2-ring, so {\mathfrak{P}}_2\subseteq {\mathfrak{P}}_c(T) and \left({\mathfrak{P}}_1+{\mathfrak{P}}_2\right)\subseteq {\mathfrak{P}}_c(T), namely \mathfrak{P}\subseteq {\mathfrak{P}}_c(T). Therefore, {\mathfrak{P}}_c(T)=\left(\begin{array}{cc}{\mathfrak{P}}_c(R)& N\\ M& {\mathfrak{P}}_c(S)\end{array}\right).

In [6], Ferrero and Puczyłowski introduced an important radical, namely the (right) singular radical \phi =\{R|R cannot be homomorphically mapped onto a nonzero semiprime non-singular ring}. It is the upper radical determined by the class of semiprime non-singular rings, and \phi \supseteq P. A radical R is called an N-radical [5] if R contains all nilpotent rings and is a left hereditary and strong radical. The prime radical P, the locally nilpotent radical L, the Jacobson radical J, and the singular radical \phi are all N-radicals.

Lemma 2.1　[5, Theorem 1] Let \mathrm{\Re } be an N-radical and R be the sum of two subrings R_{\text{1}},R_{\text{2}}. If R_1 is nilpotent and R_2\in \mathrm{\Re }, then R\in \mathrm{\Re }.

Theorem 2.2　 Let \mathrm{\Re } be any N-radical. It is supposed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq \mathrm{\Re }(R),MN\subseteq \mathrm{\Re }(S). Then the \mathrm{\Re }-radical of the Morita context ring T=(\genfrac{}{}{0ex}{}{RN}{MS}) is

\mathrm{\Re }(T)=\left(\begin{array}{cc}\mathrm{\Re }(R)& N\\ M& \mathrm{\Re }(S)\end{array}\right).

Proof　Let

T^{\prime }=\left[\left(\begin{array}{cc}0& N\\ 0& 0\end{array}\right)+\left(\begin{array}{cc}\mathrm{\Re }(R)& 0\\ 0& \mathrm{\Re }(S)\end{array}\right)\right]+\left(\begin{array}{cc}0& 0\\ M& 0\end{array}\right)=T_1+T_2,

where

T_1=\left(\begin{array}{cc}0& N\\ 0& 0\end{array}\right)+\left(\begin{array}{cc}\mathrm{\Re }(R)& 0\\ 0& \mathrm{\Re }(S)\end{array}\right)=T_3+T_4.

Since T_3=\left(\begin{array}{cc}0& N\\ 0& 0\end{array}\right) is a nilpotent ring and T_4=\left(\begin{array}{cc}\mathrm{\Re }(R)& 0\\ 0& \mathrm{\Re }(S)\end{array}\right)\cong \mathrm{\Re }(R)\oplus \mathrm{\Re }(S) is a \mathrm{\Re }-radical ring, according to Lemma 2.1, T_{\text{1}} is a \mathrm{\Re }-radical ring and T_2=\left(\begin{array}{cc}0& 0\\ M& 0\end{array}\right) is nilpotent. Based on Lemma 2.1, T^{\prime }=T_1+T_2 is a \mathrm{\Re }-radical ring. However, T^{\prime } is an ideal of T, so T^{\prime }\subseteq \mathrm{\Re }(T). On the other hand, T/T^{\prime }\cong R/\mathrm{\Re }(R)\oplus S/\mathrm{\Re }(S) is \mathrm{\Re }\text{-} semisimple, so T\subseteq \mathrm{\Re }\left(T^{\prime }\right). Therefore, \mathrm{\Re }(T)=\left(\begin{array}{cc}\mathrm{\Re }(R)& N\\ M& \mathrm{\Re }(S)\end{array}\right).

Corollary 2.1　 It is supposed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq \phi (R),MN\subseteq \phi (S). Then

(1) the singular radiccal of T is

\phi (T)=\left(\begin{array}{cc}\phi (R)& N\\ M& \phi (S)\end{array}\right);

(2) T is a semiprime singular ring \iff RandS are semiprime singular rings.

Proof　(1) can be obtained from Theorem 2.2.

(2) According to [9, Theorem 1.14],

T is semiprime singular \iff T is \phi-semisimple \iff \phi (T)=0

\begin{array}{rl}& \iff \phi (R)=0,\phi (S)=0,N=0,M=0\\ & \iff R,S\text{are semiprime singular rings and}N=0,M=0.\end{array}

The proof is complete.

Corollary 2.2　(1) It is supposed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq P(R),MN\subseteq P(S). Then the prime radical of T is

P(T)=\left(\begin{array}{cc}P(R)& N\\ M& P(S)\end{array}\right).

(2) It is assumed that the trace ideals NM and MN of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) are nilpotent. Then T is a 2-prime ring if and only if R,S are 2-prime rings.

(3) It is assumed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq L(R),MN\subseteq L(S). Then the locally nilpotent radical of T is

L(T)=\left(\begin{array}{cc}L(R)& N\\ M& L(S)\end{array}\right).

Proof　(1) and (3) can be obtained from Theorem 2.2.

(2) According to [3, Proposition 2.1], T is a 2-prime ring if and only if P(T)=P_2(T). By (1) and Theorem 1.1, it holds if and only if P(R)=P_2(R) and P(S)=P_2(S), namely if and only if R,S are 2-prime rings.

A ring R is called a Jacobson ring if every prime ideal of R is an intersection of maximal left ideals of R. Equivalently, J(R)=P(R).

Corollary 2.3　 It is supposed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq J(R),MN\subseteq J(S). Then

(1) the Jacbson radical of T is

J(T)=\left(\begin{array}{cc}J(R)& N\\ M& J(S)\end{array}\right);

(2) T is a Jacobson ring if and only if R,S are Jacobson rings.

Beidar et al. [2] introduced an Amitsur-Kurosh root {\mathfrak{P}}_e, called the Behrens radical, and proved that a ring R was a Behrens radical ring if and only if every left ideal of R was a Brown-McCoy radical ring (see [2, Proposition 3.1]). Moreover, .

Theorem 2.3　 It is assumed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM\subseteq J(R),MN\subseteq J(S). Then the Behrens radical of T is

{\mathfrak{P}}_e(T)=\left(\begin{array}{cc}{\mathfrak{P}}_e(R)& N\\ M& {\mathfrak{P}}_e(S)\end{array}\right).

Proof　Let T_0=\left(\begin{array}{cc}{\mathfrak{P}}_e(R)& N\\ M& {\mathfrak{P}}_e(S)\end{array}\right). Since T/T_0\cong R/{\mathfrak{P}}_e(R)\oplus S/{\mathfrak{P}}_e(S) is {\mathfrak{P}}_e-semisimple, then {\mathfrak{P}}_e(T)\supseteq T_0. For the ideal T_0 of T, any ideal of T_0 has the form L=\left(\begin{array}{cc}I& N_1\\ M_1& J\end{array}\right), where I,J are left ideals of {\mathfrak{P}}_e(R),{\mathfrak{P}}_e(S) respectively. By Theorem 1.2 (2), L is a Brown-McCoy radical ring \iff I,J are Brown-McCoy radical rings. It shows that every left ideal of T_0 is a Brown-McCoy radical ring. Therefore, according to [2, Proposition 3.1], T_0 is a Behrens radical ring and T_0\subseteq {\mathfrak{P}}_e(T).

A ring R is called strongly regular, if for every a of R, there exists t\in R such that a=a^{2}t. The class of strongly regular rings is a radical class. The strongly regular radical G(R) of R is the only largest strongly regular ideal of R [7].

Theorem 2.4　 For any Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]), there is

G(T)=\left(\begin{array}{cc}G(R)& 0\\ 0& G(S)\end{array}\right).

Proof　According to [18, Theorem 1.3], T_{\alpha }=\left(\begin{array}{cc}{A}_{\alpha }& V_{\alpha }\\ W_{\alpha }& B_{\alpha }\end{array}\right) is a G-ideal of T if and only if

G\left(A_{\alpha }\right)=A_{\alpha },G\left(B_{\alpha }\right)=B_{\alpha },V_{\alpha }=0,W_{\alpha }=0.

Therefore

\begin{array}{rl}G(T)& =\sum _{\alpha }{T}_{\alpha }\left(T_{\alpha }\text{runs over all}G\text{-ideals of}T\right)\\ & =\sum _{\alpha }\left(\begin{array}{cc}{A}_{\alpha }& 0\\ 0& B_{\alpha }\end{array}\right)\left(A_{\alpha },B_{\alpha }\text{run over all}G\text{-ideals of}R\text{and}S\text{respectively}\right)\\ & =\left(\begin{array}{cc}G(R)& 0\\ 0& G(S)\end{array}\right).\end{array}

The proof is completed.

Corollary 2.4　 It is assumed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM=0,MN=0. Then T is R-semisimple if and only if R,S are R-semisimple and N=0,M=0, where R represents all the radicals mentioned above.

A ring R is called an NI-ring [10] if its nil radical K(R)=\mathrm{N}\mathrm{i}\mathrm{l}(R). A ring R is called a J-ring [12] if J(R)=\mathrm{N}\mathrm{i}\mathrm{l}(R).

Corollary 2.5　 It is supposed that the trace ideals of the Morita context (R,S,M,N,(\cdot ,\cdot ),[\cdot ,\cdot ]) satisfy NM=0,MN=0. Then

(1) T is an NI-ring if and only if R,S are NI-rings.

(2) T is a J-ring if and only if R,S are J-rings.

Proof　It firstly requires to proof

Nil(T)=\left(\begin{array}{cc}Nil(R)& N\\ M& Nil(S)\end{array}\right).

It is known that C=\left(\begin{array}{cc}Nil(R)& N\\ M& Nil(S)\end{array}\right) is a subset of nilpotent elements of T, so C\subseteq \mathrm{N}\mathrm{i}\mathrm{l}(T). From {(\genfrac{}{}{0ex}{}{av}{wb})}^n=0, there are a^n=0, b^n=0 and\left(\begin{array}{cc}a& v\\ w& b\end{array}\right)\in C. As a result, C\supseteq Nil(T).

(1) T is an NI-ring \iff K(T)=Nil(T)\iff K(R)=Nil(R),K(S)=Nil(S)\iff R,S are NI-rings.

(2) The proof is similar to (1).

Example 2.1　(1) Let R,S be two rings, and {}_R{N}_S be a bimodule. The upper triangular matrix ring \left(\begin{array}{cc}R& R\\ 0& R\end{array}\right) and the zero product extension ring \left(\begin{array}{cc}R& N\\ 0& S\end{array}\right) are Morita context rings with trace ideals equal to 0;

(2) let V be an n-dimensional vector space over the field K,

R=\mathrm{\Lambda }(V)=K\oplus {\mathrm{\Lambda }}^1(V)\oplus {\mathrm{\Lambda }}^2(V)\oplus \cdots \oplus {\mathrm{\Lambda }}^n

be the exterior algebra of V, and m={\mathrm{\Lambda }}^1(V)\oplus {\mathrm{\Lambda }}^2(V)\oplus \cdots \oplus {\mathrm{\Lambda }}^n(V). Then m is an ideal of R, and m^{n+1}=0, J(R)=m. The trace ideals Rm and mR of the Morita context (R,m,R,R) are nilpotent;

(3) let A_{x\times y} be the set of x -by- y matrices over the ring A. Then the Morita context ring

\left(\begin{array}{cc}{R}_{n\times n}& J(R{)}_{n\times m}\\ R_{m\times n}& R_{m\times m}\end{array}\right)

satisfies J(R{)}_{n\times m}{R}_{m\times n}\subseteq J\left(R_{n\times n}\right) and R_{m\times n}J(R{)}_{n\times m}\subseteq J\left(R_{m\times m}\right)．

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