Quasipolar Matrix Rings Over Local Rings

Abstract

Abstract. A ring R is called quasipolar if for every a ∈ R there exists p 2 = p ∈ R such that p ∈ comm 2 R (a), a + p ∈ U (R) and ap ∈ R qnil . The class of quasipolar rings lies properly between the class of strongly π-regular rings and the class of strongly clean rings. In this paper, we determine when a 2 × 2 matrix over a local ring is quasipolar. Necessary and sufficient conditions for a 2 × 2 matrix ring to be quasipolar are obtained.

Certain strongly clean matrices over local rings

“…In a ring R , evidently, { elements having pseudo Drazin inverses } ⊆ { quasipolar elements } ⊆ { strongly clean elements }. The subjects of strongly clean rings, quasipolar rings, and pseudo Drazin inverses were extensively studied in [1][2][3][4][5]7] and [10][11][12].…”

On ps-Drazin inverses in a ring

“…, where α ∈ 1 + J(R) and β ∈ J(R ) ∈ M 2 (Z (2) ) . Then A ∈ M 2 (Z (2) ) has a ps-Drazin inverse, but B does not.…”

“…Proof Clearly, Z (2) is a commutative local ring with J(Z (2) ) = 2Z (2) . As tr(A) = 7 and det(A) = −8, we see that the equation x 2 − tr(A)x + det(A) = 0 has a root −1 ∈ 1 + J(Z (2) ) and a root 8 ∈ J(Z (2) ). According to…”

“…A ∈ M 2 (Z(2) ) has a ps-Drazin inverse. Clearly,B 2 , (I 2 − B) 2 ̸ ∈ M 2 (J(R)).In light of Lemma 5.1, conditions (1) and (2) in Corollary 5.5 are not satisfied.…”

“…We take σ : Z (2) → Q to be the natural map, and p(x) = x 2 − 5x − 2 ∈ Z (2) [x] . Clearly, p * (x) = x 2 − 5x − 2 ∈ Q[x]is irreducible, and so x 2 − 5x − 2 = 0 is not solvable in Z(2) . By virtue of Corollary 5.5, B ∈ M 2 (Z (2) ) does not have a ps-Drazin inverse.2…”