Frontiers of Mathematics in China >
Zero divisors and prime elements of bounded semirings
Received date: 25 Dec 2011
Accepted date: 31 Jul 2014
Published date: 29 Oct 2014
Copyright
A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z(A) of nonzero zero divisors of A is A \ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z(A) = A \ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either |Z(A)| = 1 or |Z(A)| = 2 and Z(A)2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset II(R) is pure and shellable, where II(R) consists of all ideals of R.
Key words： Bounded semiring; zero divisor; prime element; small Z(A); ideal structure of ring
Tongsuo WU , Yuanlin LI , Dancheng LU . Zero divisors and prime elements of bounded semirings[J]. Frontiers of Mathematics in China, 2014 , 9(6) : 1381 -1399 . DOI: 10.1007/s11464-014-0423-1
| 1 |
Aalipour G, Akbari S, Behboodi M, Nikandish R, Nikmehr M J, Shaveisi F. The classification of the annihilating-ideal graph of a commutative ring. Algebra Colloq, 2014, 21(2): 249-256
|
| 2 |
Aalipour G, Akbari S, Nikandish R, Nikmehr M J, Shaveisi F. Minimal prime ideals and cycles in annihilating-ideal graphs. Rocky Mountain J Math, 2013, 43(5): 1415-1425
|
| 3 |
Anderson D F, Livingston P S. The zero-divisor graph of a commutative ring. J Algebra, 1999, 217: 434-447
|
| 4 |
Atiyah M F, MacDonald I G. Introduction to Commutative Algebra. Reading: Addison- Wesley, 1969
|
| 5 |
Behboodi M, Rakeei Z. The annihilating-ideal graph of commutative rings I. J Algebra Appl, 2011, 10(4): 727-739
|
| 6 |
Bistarelli S, Montanari U, Rossi F. Semiring-based constraint satisfaction and optimization. J ACM, 1997, 44(2): 201-236
|
| 7 |
Bistarelli S, Montanari U, Rossi F, Schiex T, Verfaillie G, Fargier H. Semiring-based CSPs and valued CSPs: frameworks, properties, and comparison. Constraints, 1999, 4(3): 199-240
|
| 8 |
Björner A. Shellable and Cohen-Macaulay partially ordered sets. Trans Amer Math Soc, 1980, 260: 159-183
|
| 9 |
DeMeyer F R, McKenzie T, Schneider K. The zero-divisor graph of a commutative semigroup. Semigroup Forum, 2002, 65: 206-214
|
| 10 |
Lam T Y. A First Course in Non-Commutative Rings. New York: Springer-Verlag, 1991
|
| 11 |
Lu D C, Wu T S. On the normal ideals of exchange rings. Sib Math J, 2008, 49: 663-668
|
| 12 |
McLean K R. Commutative Artinian principal ideal rings. Proc Lond Math Soc, 1973, 26(3): 249-272
|
| 13 |
Wu T S, Lu D C. Sub-semigroups determined by the zero-divisor graph. Discrete Math, 2008, 308: 5122-5135
|
| 14 |
Wu T S, Yu H Y, and Lu D C. The structure of finite local principal ideal rings. Comm Algebra, 2012, 40(12): 4727-4738
|
| 15 |
Yu H Y, Wu T S. On realizing zero-divisor graphs of bounded semirings. Preprint
|
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