Yosum Kurtulmaz

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.

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].