H. Kose

On feckly clean rings

A ring R is feckly clean provided that for any a ∈ R there exists an element e ∈ R and a full element u ∈ R such that a = e + u, eR(1 - e) ⊆ J(R). We prove that a ring R is feckly clean if and only if for any a ∈ R, there exists an element e ∈ R such that V(a) ⊆ V(e), V(1 - a) ⊆ V(1 - e) and eR(1 - e) ⊆ J(R), if and only if for any distinct maximal ideals M and N, there exists an element e ∈ R such that e ∈ M, 1 - e ∈ N and eR(1 - e) ⊆ J(R), if and only if J-spec(R) is strongly zero-dimensional, if and only if Max(R) is strongly zero-dimensional and every prime ideal containing J(R) is contained in a unique maximal ideal. More explicit characterizations are also discussed for commutative feckly clean rings.

STRONGLY CLEAN MATRICES OVER COMMUTATIVE LOCAL RINGS

An element of a ring is called strongly clean provided that it can be written as the sum of an idempotent and a unit that commute. We characterize, in this paper, the strongly cleanness of matrices over commutative local rings. This partially extend many known results such as Theorem 12 in Borooah, Diesl and Dorsey [Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra212 (2008) 281–296], Theorem 3.2.7 and Proposition 3.3.6 in Dorsey [Cleanness and strong cleanness of rings of matrices, Ph.D. thesis, University of California, Berkeley (2006)], Theorem 2.3.14 in Fan [Algebraic analysis of some strongly clean and their generalization, Ph.D. thesis, Memorial University of Newfoundland, Newfoundland (2009)], Theorem 3.1.9 and Theorem 3.1.26 in Yang [Strongly clean rings and g(x)-clean rings, Ph.D. thesis, Memorial University of Newfoundland, Newfoundland (2007)].

Factorizations of Matrices over Projective-free Rings

An element of a ring R is called strongly J#-clean provided that it can be written as the sum of an idempotent and an element in J#(R) that commute. In this paper, we characterize the strong J#-cleanness of matrices over projective-free rings. This extends many known results on strongly clean matrices over commutative local rings.

Strongly Clean Matrices Over Power Series

An n×n matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let A(x) ∈ Mn ( R[[x]] ) . We prove, in this note, that A(x) ∈ Mn ( R[[x]] ) is strongly clean if and only if A(0) ∈ Mn(R) is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.

EXTENSIONS OF STRONGLY π-REGULAR RINGS

An ideal I of a ring R is strongly π-regular if for any x ∈ I there exist n ∈ N and y ∈ I such that x = xy. We prove that every strongly π-regular ideal of a ring is a B-ideal. An ideal I is periodic provided that for any x ∈ I there exist two distinct m,n ∈ N such that x = x. Furthermore, we prove that an ideal I of a ring R is periodic if and only if I is strongly π-regular and for any u ∈ U(I), u−1 ∈ Z[u].