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Classes of Commutative Clean Rings

Wolf Iberkleid, Warren Wm. McGovern (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Let A be a commutative ring with identity and I an ideal of A . A is said to be I - c l e a n if for every element a ∈ A there is an idempotent e = e 2 ∈ A such that a - e is a unit and a e belongs to I . A filter of ideals, say ℱ , of A is Noetherian if for each I ∈ ℱ there is a finitely generated ideal J ∈ ℱ such that J ⊆ I . We characterize I -clean rings for the ideals 0 , n ( A ) , J ( A ) , and A , in terms of the frame of multiplicative Noetherian filters of ideals of A , as well as in terms of more classical ring properties.

Conditions under which R ( x ) and R 〈 x 〉 are almost Q-rings

Hani A. Khashan, H. Al-Ezeh (2007)

Archivum Mathematicum

All rings considered in this paper are assumed to be commutative with identities. A ring R is a Q -ring if every ideal of R is a finite product of primary ideals. An almost Q -ring is a ring whose localization at every prime ideal is a Q -ring. In this paper, we first prove that the statements, R is an almost Z P I -ring and R [ x ] is an almost Q -ring are equivalent for any ring R . Then we prove that under the condition that every prime ideal of R ( x ) is an extension of a prime ideal of R , the ring R is a (an almost)...

Congruences and ideals in ternary rings

Ivan Chajda, Radomír Halaš, František Machala (1997)

Czechoslovak Mathematical Journal

A ternary ring is an algebraic structure ℛ = ( R ; t , 0 , 1 ) of type ( 3 , 0 , 0 ) satisfying the identities t ( 0 , x , y ) = y = t ( x , 0 , y ) and t ( 1 , x , 0 ) = x = ( x , 1 , 0 ) where, moreover, for any a , b , c ∈ R there exists a unique d ∈ R with t ( a , b , d ) = c . A congruence θ on ℛ is called normal if ℛ / θ is a ternary ring again. We describe basic properties of the lattice of all normal congruences on ℛ and establish connections between ideals (introduced earlier by the third author) and congruence kernels.

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