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A unit-picker is a map that associates to every ring R a well-defined set (R) of central units in R which contains 1 R , is invariant under isomorphisms of rings, is closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let be a unit-picker. A ring R is called U U if U (R) = (R) + nil (R) and U J if U (R) = (R) + J (R) and U N J if U (R) = (R) + nil (R) + J (R). An extensive study of these rings is conducted, and their connections with strongly nil -clean rings and semi -Boolean rings are investigated. When is specified, known results of U U -rings, U J -rings and U N J -rings are obtained and new results are proved. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, m>1 an integer and q>1 an odd number. The following questions are addressed: (1) when is every element in M n (R) and, respectively, in T n (R) a product of q-potents? (2) when is every element in M n (R) and, respectively, in T n (R) a product of mq-potents? and (3) when is every element in M n (R) and, respectively, in T n (R) a product of an idempotent and a q-potent? [ABSTRACT FROM AUTHOR]

A unit-picker is a map that associates to every ring R a well-defined set (R) of central units in R which contains 1 R and is invariant under isomorphisms of rings and closed under taking inverses, and which satisfies certain set containment conditions for quotient rings, corner rings and matrix rings. Let be a unit-picker. A ring R is called (strongly) nil -clean if for each a ∈ R , a = v e + b where v ∈ (R) , e 2 = e ∈ R and b ∈ R is nilpotent (and a b = b a). An extensive study of (strongly) nil -clean rings is conducted. When is specified, known results of the much-studied (strongly) nil-clean rings and weakly nil-clean rings are re-proved and new results of other nil-clean-like rings are obtained. [ABSTRACT FROM AUTHOR]

A ∗ -ring R is called a nil ∗ -clean ring if every element of R is a sum of a projection and a nilpotent. Nil ∗ -clean rings are the ∗ -version of nil-clean rings introduced by Diesl. This paper is about the nil ∗ -clean property of rings with emphasis on matrix rings. We show that a ∗ -ring R is nil ∗ -clean if and only if J (R) is nil and R / J (R) is nil ∗ -clean. For a 2-primal ∗ -ring R , with the induced involution given by (a i j ) ∗ = (a i j ∗) T , the nil ∗ -clean property of M n (R) is completely reduced to that of M n (Z 2). Consequently, M n (R) is not a nil ∗ -clean ring for n = 3 , 4 , and M 2 (R) is a nil ∗ -clean ring if and only if J (R) is nil, R / J (R) is a Boolean ring and a ∗ − a ∈ J (R) for all a ∈ R. [ABSTRACT FROM AUTHOR]

A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units. In this note, we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property. [ABSTRACT FROM AUTHOR]

Every matrix in the triangular matrix ringover a bleached local ringRis similar to a ‘simple form’, which is, in most cases, contained in a subringofwithisomorphic to a direct product of two triangular matrix rings overRof smaller size. This result, called the embedding theorem, suggests a new approach for handling triangular matrix rings. It is applied to proving several results on the strong cleanness and strong 2-sum property of triangular matrix rings. [ABSTRACT FROM AUTHOR]

Generalizing the notion of nil-cleanness from [A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197-211], in parallel to [P. V. Danchev and W. Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra 425 (2015) 410-422], we define the concept of weak nil-cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean. [ABSTRACT FROM AUTHOR]

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring $$\mathbb{M}_n (R)$$ is weakly nil-clean, and to show that the endomorphism ring End( V) over a vector space V is weakly nil-clean if and only if it is nil-clean or dim( V) = 1 with D≅= ℤ. [ABSTRACT FROM AUTHOR]

The article concerns the question of when a generalized matrix ringKs(R) over a local ringRis quasipolar. For a commutative local ringR, it is proved thatKs(R) is quasipolar if and only if it is strongly clean. For a general local ringR, some partial answers to the question are obtained. There exist noncommutative local ringsRsuch thatKs(R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix ofKs(R) (whereRis a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5] are improved or extended. [ABSTRACT FROM PUBLISHER]

This article concerns mainly on various ring properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a generalized matrix ring over a ring to satisfy a certain ring property which is among being semilocal, semiperfect, left perfect, semiprimary, semipotent, potent, clean, strongly π-regular, semiregular, etc. Many known results on a formal triangular matrix ring are extended to a Morita context or a trivial Morita context. Some questions on this subject raised by Varadarajan in [22] are answered. [ABSTRACT FROM PUBLISHER]

Abstract: This paper concerns the formal matrix ring over a ring R defined by a central element s in R. Various basic properties of these rings are established; the isomorphism problem between these rings is addressed; some known results on the usual matrix rings are extended to the formal matrix rings; and the Cayley–Hamilton Theorem is proved for matrices in a formal matrix ring. [Copyright &y& Elsevier]

A ring R is said to satisfy the Goodearl-Menal condition if for any x, y ∈ R, there exists a unit u of R such that both x -- u and v --u-1 are units of R. It is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian, then R satisfies the Goodearl-Menal condition if and only if no homomorphic image of R is isomorphic to either Z2 or Z3 or M2(Z2). These results correct two existing results. Some consequences are discussed. [ABSTRACT FROM AUTHOR]

Abstract: Let K s(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K s(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K s(R) to be strongly clean and, respectively, for the ring K s(R) to be strongly clean. Specializing to s =1 yields some known results. New families of strongly clean rings are presented. [Copyright &y& Elsevier]

Abstract: A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [G. Borooah, A.J. Diesl, T.J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (1) (2008) 281–296] completely characterized the commutative local rings R for which is strongly clean. For a general local ring R and , however, it is unknown when the matrix ring is strongly clean. Here we completely characterize the local rings R for which is strongly clean. [Copyright &y& Elsevier]

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean. [ABSTRACT FROM AUTHOR]

A ring R is called left P-injective if for every a ∈ R , aR = r(l( a )) where l( ⋅ ) and r( ⋅ ) denote left and right annihilators respectively. The ring R is called left GP -injective if for any 0 ≠ a ∈ R , there exists n > 0 such that a n ≠ 0 and a n R = r(l( a n )). As a response to an open question on GP -injective rings, an example of a left GP -injective ring which is not left P -injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 n ( R ) is left GP -injective. [ABSTRACT FROM AUTHOR]

A ring R is called Armendariz if, whenever (∑si=0aixi(∑tj=0bjxj) = 0 in R[x], aibj = 0 for all i and j. In this paper, some "relatively maximal" Armendariz subrings of matrix rings are identified, and a necessary and sufficient condition for a trivial extension to be Armendariz is obtained. Consequently, new families of Armendariz rings are presented. [ABSTRACT FROM AUTHOR]

Abstract: A ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1], we prove that the matrix ring over a division ring D is a nil-clean ring if and only if . As consequences, it is shown that the matrix ring over a strongly regular ring R is a nil-clean ring if and only if R is a Boolean ring, and that a semilocal ring R is nil-clean if and only if its Jacobson radical is nil and is a direct product of matrix rings over . [Copyright &y& Elsevier]

Abstract: A ring is called uniquely clean if every element is uniquely the sum of an idempotent and a unit. The rings described by the title include uniquely clean rings, and they arise as triangular matrix rings over commutative uniquely clean rings. Various basic properties of these rings are proved and many examples are given. [Copyright &y& Elsevier]

Abstract: A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local rings are strongly clean. It is unknown when a matrix ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005] that for any prime number p, the matrix ring is strongly clean where is the ring of p-adic integers, but is not strongly clean where is the localization of at the prime ideal generated by p. Let R be a commutative local ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for to be strongly clean. As consequences, is strongly clean iff is strongly clean iff is strongly clean iff is strongly clean. [Copyright &y& Elsevier]