Rings whose non-invertible elements are strongly weakly nil-clean

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.

It is proved that for matrices [Formula: see text], [Formula: see text] in the [Formula: see text] by [Formula: see text] upper triangular matrix ring [Formula: see text] over a domain [Formula: see text], if [Formula: see text] is nonzero and central in [Formula: see text] then [Formula: see text]. The [Formula: see text] by [Formula: see text] full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

We systematically study those rings whose nonunits are a sum of an idempotent and a nilpotent. Some crucial characteristic properties are completely described as well as some structural results for this class of rings are obtained. This work somewhat continues two publications on the subject due to A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211 and A. Karimi–Mansoub, T. Kosan and Y. Zhou, Rings in which every unit is a sum of a nilpotent and an idempotent, Contemp. Math. 715 (2018) 189–203.

Group Gradings on Full Matrix 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.

Ring of classes and character ring of a finite group

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.

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.

For a unital Banach algebra A over C, we give improvements on the well-known multiplicative spectral characterization of the Jacobson radical, (1), in terms of several spectral parameters. In particular, we show that every non-invertible element a ∈ A for which the number of elements in the spectrum of ax is less or equal to the number of elements in the spectrum of x, for all x in an arbitrary small neighborhood of the unit, must belong to the Jacobson radical.

It is shown that if F is a field of characteristic zero and G is a group such that the group ring F[G] is semilocal then G must be finite.A generalization to group rings over rings is given.A ring R is semilocal if R/J(R) is artinian, where I(R) denotes the Jacobson radical of R. R is said to be local if R/I(R) is a division ring.It is well known that the group ring is never local for a field of characteristic zero unless the group is trivial.As it is conjectured that J(F[G]) = (0) whenever Fis a field of characteristic zero, we expect F[G] semilocal to imply G finite, and this is easily proved if F is not algebraic over the rationals, by a theorem of Amitsur [1].The result in this paper may be interpreted as saying that the radical of the rational group ring cannot be "too large".We would like to thank D. S. Passman for his helpful suggestions.Lemma 1.Let K be a central subfield of a division ring D and let T G Mn(D), the full n by n matrix ring over D. Then the set S (T) = {k G K: 1 -k~ T is singular] has at most n elements.Proof.When written on the right, the elements of Mn(D) may be regarded as left Z)-linear transformations from the vector space D" to D".If 1 -k~] T is singular, there is a nonzero vector v G D" such that v(\ -k~ T) = 0, or vT = kv since k commutes with v. Hence k is an eigenvalue of T. Standard arguments of linear algebra show that eigenvectors in D" corresponding to distinct eigenvalues of T in the centre of D are Z)-linearly independent. Corollary.Let R be a completely reducible K-algebra and let x G R. Then the set S(x) = {k G K: 1 -k~ x is not a unit in R] is finite.The following clever lemma forms a major part of the proof of the theorem in [3],Lemma 2 (Formanek).Let K be a subfield of the reals and let x = 2"i aS

This paper studies the question, if $R$ is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over $R$? Theorem 2.6. If $R$ is an algebra over a field with at least $q$ elements, and $R$ satisfies ${x^q} = 0$, and $G$ is a group with an abelian subgroup of index $k$, then the group ring $R(G)$ satisfies ${x^t} = 0$, where $t = q{k^2} + 2$. Theorem 3.2. If $R$ is a ring satisfying a standard identity, and $G$ is a finite group, then $R(G)$ satisfies a standard identity. Theorem 3.4. If $R$ is an algebra over a field, and $R$ satisfies a standard identity, then the $k$-by-$k$ matrix ring ${R_k}$ satisfies a standard identity. Each theorem specifies the degree of the polynomial identity.

Commutative weakly nil clean unital rings

A new construction method for codes using encodings from group rings is presented. They consist primarily of two types, zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; e.g. cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals. Using an isomorphism between group rings and a certain well- defined ring of matrices, equivalent matrix codes are established with resulting generator and check matrices. Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties may more easily be expressed in terms of group ring properties.

A radical R, in the sense of Kurosh--Amitsur, is said to be compact if, given any collection of radicals X such that R ≤;VX, we have R ≤;VX' for some finite subcollection X' of X. A ring A is said to be radical compact if the lower radical on the singleton {A} is compact. This paper explores the relationship between radical compact rings and rings satisfying certain finiteness conditions. Closure properties of the class of all radical compact rings are also investigated.