Huanyin Chen

On Strongly J-Clean Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J-clean in case each of its elements is strongly J-clean. We investigate, in this article, strongly J-clean rings and ultimately deduce strong J-cleanness of T n (R) for a large class of local rings R. Further, we prove that the ring of all 2 × 2 matrices over commutative local rings is not strongly J-clean. For local rings, we get criteria on strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices. The strong J-cleanness of a 2 × 2 matrix over commutative local rings is completely characterized by means of a quadratic equation.

On Uniquely Clean Rings

A ring R is uniquely (nil) clean in case for any a ∈ R there exists a unique idempotent e ∈ R such that a − e ∈ R is invertible (nilpotent). We prove in this article that a ring R is uniquely clean if and only if R is an exchange ring with all idempotents central, and that for all maximal ideals M of R, R/M ≅ ℤ2. It is shown that every uniquely clean ring of which every prime ideal of R is maximal is uniquely nil clean. Further, we prove that a ring R is uniquely nil clean if and only if R/J(R) is Boolean and R is a π-regular ring with all idempotents central.

On Strongly Nil Clean Matrices

An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. We characterize, in this article, the strongly nil cleanness of matrices over projective-free rings in terms of the factorizations of their characteristic polynomials.

Strongly 2-nil-clean rings

A ring R is strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such rings are obtained. We prove that a ring R is strongly 2-nil-clean if and only if for all a ∈ R, a − a3 ∈ R is nilpotent, if and only if for all a ∈ R, a2 ∈ R is strongly nil-clean, if and only if every element in R is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring R is strongly 2-nil-clean if and only if R/J(R) is tripotent and J(R) is nil, if and only if R≅R1,R2 or R1 × R2, where R1/J(R1) is a Boolean ring and J(R1) is nil; R2/J(R2) is a Yaqub ring and J(R2) is nil. Strongly 2-nil-clean group algebras are investigated as well.

STRONGLY J-CLEAN MATRICES OVER LOCAL RINGS

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8, Theorems 2.7 and 3.2], [9, Theorem 2.6], and [11, Theorem 7].

On strongly nil clean rings

ABSTRACT A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a ring R is strongly nil clean if and only if for any a ∈ R, there exists an idempotent e ∈ ℤ[a] such that a − e ∈ N(R), if and only if R is periodic and R∕J(R) is Boolean, if and only if each prime factor ring of R is strongly nil clean. Further, we prove that R is strongly nil clean if and only if for all a ∈ R, there exist n ∈ ℕ,k ≥ 0 (depending on a) such that , if and only if for fixed m,n ∈ ℕ, for all a ∈ R. These also extend known theorems, e.g, [5, Theorem 3.21], [6, Theorem 3], [7, Theorem 2.7] and [12, Theorem 2].

Diagonal matrix reduction over Hermite rings

ABSTRACT A commutative ring R is an elementary divisor ring if every matrix over R admits a diagonal reduction. In this paper, we define the term ‘Zabvasky subset’ of a ring to study diagonal matrix reduction. Let S be a Zabavsky subset of a Hermite ring R. We prove that R is an elementary divisor ring if and only if with implies that there exist such that . If with implies that there exists a such that , then R is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.

Strongly Weakly Nil-clean Rings

A ring R is strongly weakly nil-clean if every element in R is the sum or difference of a nilpotent and an idempotent that commutes. We prove, in this paper, that a ring R is strongly weakly nil-cl...

ELEMENTARY MATRIX REDUCTION OVER ZABAVSKY RINGS

We prove, in this note, that a Zabavsky ring R is an elementary divisor ring if and only if R is a ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [7, Theorem 4], [9, Theorem 1.2.14], [11, Theorem 4] and [12, Theorem 7].

Cancellation of Small Projectives Over Exchange Rings

An ideal I of an exchange ring R is strongly separative provided that for any A,B ∈ FP(R), A⊕ A≅ A⊕ B ↠ A≅ B. If I is a strongly separative ideal of an exchange ring R, then each a ∈ R satisfying is unit-regular. The converse is true for regular ideals. Furthermore, we prove that strong separativity for such regular ideals can be determined only by one-sided units. Clean property of elements in strongly separative ideals of exchange rings is also studied.