Remarks on Centers of Rings

For a ring endomorphism [Formula: see text], a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, [Formula: see text]-[Formula: see text]), is introduced and studied. It is shown that the [Formula: see text]-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

A $\ast$-ring
 $R$ is called a $\pi$-Baer $\ast$-ring, if for any projection invariant left ideal $Y$ of $R$, the right annihilator of $Y $
 is generated, as a right ideal, by a projection.
 In this note, we
 study some properties of such $\ast$-rings.
 We indicate interrelationships between the $\pi$-Baer $\ast$-rings and related classes of rings such as
 $\pi$-Baer rings, Baer $\ast$-rings, and quasi-Baer $\ast$-rings. We announce several
 results on $\pi$-Baer $\ast$-rings.
 We show that this notion is well-behaved with respect to
 polynomial extensions and full matrix rings.
 Examples are provided to explain and delimit our results.

Let R be a commutative ring with identity. Then we prove <TEX>$M_n(R)=GL_n(R)$</TEX> <TEX>${\cup}$</TEX>{<TEX>$A{\in}M_n(R)\;{\mid}\;detA{\neq}0$</TEX> and det <TEX>$A{\neq}U(R)$</TEX>}<TEX>${\cup}Z(M-n(R))$</TEX> where U(R) denotes the set of all units of R. In particular, it will be proved that the full matrix ring <TEX>$M_n(F)$</TEX> over a field F is the disjoint union of the general linear group <TEX>$GL_n(F)$</TEX> of degree n over the field F and the set <TEX>$Z(M_n(F))$</TEX> of all zero-divisors of <TEX>$M_n(F)$</TEX>. Using the result and universal mapping property we prove that <TEX>$M_n(F)$</TEX> is its total ring of fractions.

Group Gradings on Full Matrix Rings

Throughout this note R will denote a local Dedekind domain with a unique maximal ideal zR and the quotient ring K. Let n(^2) be an integer. As usual, an i?-order A in the full nXn matrix ring (K)n over K is an i?-subalgebra of (K)n such that A is finitely generated as an i?-module and AK=(K)n. An i?-orderin (K)n is called tiledif it contains n orthogonal idempotents. If A is a tiled Z?-order in (K)n then up to conjugation we may assume that it contains the idempotents en (l^z^n), where {etj\l which is constructed from their graph theoretic properties. In this note we shall remove the hypothesis between (xR)n and (/?), and prove the following

Let K be a commutative field and let V be an n-dimensional vector space over K. We denote by L(V) the ring of all K-linear endomorphisms of V into itself. A subring of L(V) is always assumed to contain the unit element of L(V), but it need not be a vector subspace of the K-algebra L(V). Suppose now that A is a subring of L(V). Then we may consider L(V) as a left or a right A-module. We may also consider V as a left A-module, i.e. if x ∈ V and if f ∈ A then we get the image element f(x) in V. If we choose a basis for V over K then we get the associated matrix representation of L(V). So in this way L(V) is identified with the full matrix ring Mn(K), and in this way the study of subrings of L(V) can be reduced to the study of subrings of Mn(K).

It is known that structural matrix rings pro-vide a natural passage from complete matrix rings to upper and lower triangular matrix rings, and they often explain the peculiarities regarding certain properties of complete matrix rings on the one hand and of triangular matrix rings on the other hand. In this paper the concept of a set of matrix units in a ring associated with a quasi-order relation is introduced and used to provide an internal char-acterisation of structural matrix rings.

The target of this work is to give a new insight in the theory of strongly weakly nil-clean rings, recently defined by Kosan and Zhou [On weakly nil-clean rings, Front. Math. China 11(4) (2016) 949–955] and further explored in detail by Chen–Sheibani [Strongly weakly nil-clean rings, J. Algebra Appl. 16 (2017)]. Indeed, we consider those rings whose non-units are strongly weakly nil-clean and succeed to establish that this class of rings is strongly [Formula: see text]-regular and, even something more, that it possesses a complete characterization in terms of the Jacobson radical and sections of the [Formula: see text] full matrix ring. Additionally, some extensions like Morita context rings and groups rings are also studied in this directory.

Berkani and Sarihr [Studia Math. (2001) 148: 251-257] showed that the set of all Drazin invertible elements in an algebra over a filed is a regularity in the sense of Kordula and M?ller [Studia Math. (1996) 119: 109-128]. In this paper, the above result is extended to the case of a ring. Counterexamples are provided to show that the set of all generalized Drazin invertible elements in a ring need not be a regularity in general. We determine when the set of all generalized Drazin invertible matrices in the 2 ? 2 full matrix ring over a commutative local ring is a regularity. We also give a sufficient condition for the set of all generalized Drazin invertible elements in a ring to be a regularity.

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.

Let R be an associative ring with identity 1. We describe all idempotent matrices with only zeros and ones on the diagonal in – the ring of upper triangular matrices over R (), and – the ring of infinite upper triangular matrices (indexed by ) over R. Moreover, when R is finite, we calculate the number of all idempotent matrices with only zeros and ones on the diagonal in .

In this paper, we introduce the concept of a triangular coefficient matrix ring and investigate the structure of its ideals.We then characterize the radicals of the ring R h [x]/x n for every positive integer n, where R h [x] denotes the Hurwitz polynomial ring and x n represents the ideal of this ring generated by x n .Furthermore, we explore several properties that are transferred between the base ring R and the matrix ring Hn(R), which is a proper subring of the triangular coefficient matrix ring.

A polynomial in noncommuting variables is vanishing, nil or central in a ring, R R , if its value under every substitution from R R is 0, nilpotent or a central element of R R , respectively. THEOREM. If R R has no nonvanishing multilinear nil polynomials then neither has the matrix ring R n {R_n} . THEOREM. Let R R be a ring satisfying a polynomial identity modulo its nil radical N N , and let f f be a multilinear polynomial. If f f is nil in R R then f f is vanishing in R / N R/N . Applied to the polynomial x y − y x xy - yx , this establishes the validity of a conjecture of Herstein’s, in the presence of polynomial identity. THEOREM. Let m m be a positive integer and let F F be a field containing no m m th roots of unity other than 1. If f f is a multilinear polynomial such that for some n &gt; 2 f m n &gt; 2{f^m} is central in F n {F_n} , then f f is central in F n {F_n} . This is related to the (non)existence of noncrossed products among p 2 {p^2} -dimensional central division rings.

An application of Hochschild cohomology to the moduli of subalgebras of the full matrix ring II