On two open problems about strongly clean rings

Abstract

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n 6 Q : n is odd}, then M 2 (R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.Throughout this paper all rings are associative with unit. For a ring R, let U(R) be the group of units of R, M n (R) the nxn matrix ring over R, and T n (R) the nxn upper triangular matrix ring over R, respectively. The identity matrix of M n (R) is denoted by / . Q means the field of rational numbers. A ring R is called clean if every element of R is a sum of an idempotent and a unit. The ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. It is shown by Camillo and Yu [2, Theorem 9] that every semiperfect ring is clean. Han and Nicholson [4, Corollary 1] showed that every matrix ring M n (R) over a clean ring is again clean.Nicholson asked whether every semiperfect ring is strongly clean [5, Question 5] and whether the matrix ring of a strongly clean ring is strongly clean [5, Question 3]. In this paper, we prove that if R -{m/n € Q : n is odd}, then M 2 (/2) is a semiperfect ring but not strongly clean. Thus, we answer the two questions above, both in the negative. Also we prove that every upper triangular matrix ring over the ring R is strongly clean. Thus, we obtain a new class of strongly clean rings. EXAMPLE 1. Let R -{m/n e Q : n is odd}. Then M 2 {R) is a semiperfect ring but it is not strongly clean.PROOF: Since R is a commutative local ring, it is semiperfect and strongly clean. Since semiperfect rings are Morita invariant, M 2 {R) is semiperfect. By direct computation, we find all nontrivial idempotents in the matrix ring M 2 (i?) are of the following types:I I , where a,b,c€ R and bc = a-a 2 . \c 1-a

On Strongly Clean Matrix Rings

“…For example, the matrix ring ‫ލ‬ 2 ‫ޚ(‬ (2) ) is not strongly clean. This fact was observed by Sánchez Campos [12] and by Wang and Chen [13] independently (answering two questions of Nicholson in [11]). When is the matrix ring over a strongly clean ring still strongly clean?…”

Quasipolar Matrix Rings Over Local Rings

“…Nicholson [15] proved that any strongly π-regular element is strongly clean by establishing the following results: for α ∈ end(M ), α is strongly π-regular if and only if there exist α-invariant submodules P and Q such that M = P ⊕ Q, α| P is an isomorphism and α| Q is nilpotent; and α is strongly clean if and only if there exist α-invariant submodules P and Q such that M = P ⊕ Q, α| P and (1 − α)| Q are isomorphisms. Some other notable results on strongly clean rings can be found in [1,2,3,4,13,16,17], etc.…”

“…In 2004, Wang and Chen [16] proved that there exists a commutative local ring R such that M 2 (R) is not strongly clean, which answered a question raised by Nicholson in [15]. This motivated many authors studied strong cleanness of matrix rings over local rings ([2, 3, 4, 12, 13, 17]).…”

A Survey of Strongly Clean Rings

“…Sánchez Campos [40] and Wang-Chen [42] found that M 2 (Z (2) ) is not strongly clean where Z (2) is the localization of the ring of integers Z at the prime ideal (2) = 2Z. Local rings are semiperfect, so M 2 (Z (2) ) is semiperfect but it is not strongly clean.…”