On the normal ideals of exchange rings

Lu, Dancheng; Wu, Tongsuo

doi:10.1007/s11202-008-0063-3

Cited by 1 publication

(1 citation statement)

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“…We remark that exchange rings include von Neumann regular rings, artinian rings, semilocal rings such that idempotents lift modulo their Jacobson radical. For further information on exchange rings, see [11] and the listed references.…”

Section: Introductionmentioning

confidence: 99%

Zero divisors and prime elements of bounded semirings

Li²,

Lu³

2014

Front. Math. China

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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 I(R) is pure and shellable, where I(R) consists of all ideals of R.

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Section: Introductionmentioning

confidence: 99%