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We find an expression for the Drazin inverse of a strongly π -regular element a in the form a i x j when a n = a n + k x for some nonnegative integer n and positive integer k. This extends the result by Azumaya, which is the case when n = 1 and k = 1 , and a result by Drazin, which is the case when n is arbitrary and k = 1. We give new proofs of several results in the literature. For instance, we give an easy proof of the result that for two commuting Drazin invertible elements a , b of R , then a + b is Drazin invertible if and only if so is 1 + a D b , where a D is the Drazin inverse of a. Our proof is akin to the case when a and b are invertible. [ABSTRACT FROM AUTHOR]

We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$. As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$ , $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$ , then R is an $n \times n$ matrix ring. This improves a result of Fuchs ['A characterisation result for matrix rings', Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R , we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so. [ABSTRACT FROM AUTHOR]

We study matrices over arbitrary rings that are integral over the base ring. For any ring R , let f , g , h ∈ R [ x 1 , x 2 , ... , x n ] with h = f g. We prove that for any pairwise commuting elements a 1 , a 2 , ... , a n ∈ R if g (a 1 , a 2 , ... , a n) = 0 , then h (a 1 , a 2 , ... , a n) = 0. As a corollary, it follows that for f ∈ n (S) [ x 1 , x 2 , ... , x n ] , S commutative ring, if A 1 , A 2 , ... , A n ∈ n (S) are pairwise commuting matrices such that f (A 1 , A 2 , ... , A n) = 0 , then g (A 1 , A 2 , ... , A n) = 0 where g = det (f) I. This result, which is a generalization of the Cayley–Hamilton Theorem, was proved by Phillips in 1919. For a positive integer n > 1 , we prove that if every matrix in n (R) satisfies a monic polynomial of degree n over R , then R is commutative. On the other hand, every diagonal matrix in n (R) , n > 1 , satisfies a monic polynomial of degree n over R precisely when R is a left duo ring. We prove that if every diagonal matrix in n (R) , n > 1 , is R -integral, then R is Dedekind finite. [ABSTRACT FROM AUTHOR]

We will give some glimpses into noncommutative ring theory by discussing some typical results. One of the aim is to demonstrate some basic ways in which noncommutative ring theory is different from its commutative version. [ABSTRACT FROM AUTHOR]

A clean decomposition a = e + u in a ring R (with idempotent e and unit u) is said to be special if a R ∩ e R = 0. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott–Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution. [ABSTRACT FROM AUTHOR]

We study rings in which ab nonzero idempotent implies ba is also an idempotent. We call such rings i-reversible. Besides studying the basic properties of i-reversible rings, we characterize i-reversible triangular matrix rings, i-reversible matrix rings over commutative rings and i-reversible exchange rings. [ABSTRACT FROM AUTHOR]

In 1942, McCoy proved that if a polynomial f(x) over a commutative ring R has a nonzero annihilator in R [ x ] , then it has a nonzero annihilator in R also. Later in 1957 McCoy proved that for f (x) ∈ R [ x ] , R an arbitrary ring, if f (x) R [ x ] is annihilated on the right by some nonzero polynomial in R [ x ] , then f (x) R [ x ] is also annihilated on the right by some nonzero element of R. Generalizing these results of McCoy we prove that a polynomial f(x) over an arbitrary ring R has a nonzero right annihilator in R if and only if there exists a nonzero polynomial g(x) in R [ x ] such that f (x) cg (x) = 0 for every c in the multiplicative monoid generated by the coefficients of f(x). We also show that taking c to be identity and coefficients of f(x) does not suffice. We also give a new proof of Nielsen's result that every reversible ring is right McCoy. We prove that an exchange right linearly McCoy ring is reduced modulo the Jacobson radical. We also prove that R = ( S M 0 T ) is a McCoy ring, where S and T are local rings with nonzero Jacobson radicals in which product of two non-units is zero and S M T is a bimodule such that J (S) M = 0 and MJ(T) = 0. This leads to a large class of exchange non-abelian McCoy rings. [ABSTRACT FROM AUTHOR]