Quasi-clean rings and strongly quasi-clean rings

An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed.

Notes on quasi-Frobenius rings

We give some new characterizations of quasi-Frobenius rings. Namely, we prove that for a ring R, the following statements are equivalent: (1) R is a quasi-Frobenius ring, (2) {M_{2}(R)} is right Johns and every closed left ideal of R is cyclic, (3) R is a left 2-simple injective left Kasch ring with ACC on left annihilators, (4) R is a left 2-injective semilocal ring such that {R/S_{l}} is left Goldie, (5) R is a right YJ-injective right minannihilator ring with ACC on right annihilators.

The M-intersection graph of ideals of a commutative ring

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module, and let [Formula: see text] be the set of all nontrivial ideals of [Formula: see text]. The [Formula: see text]-intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. For every multiplication [Formula: see text]-module [Formula: see text], the diameter and the girth of [Formula: see text] are determined. Among other results, we prove that if [Formula: see text] is a faithful [Formula: see text]-module and the clique number of [Formula: see text] is finite, then [Formula: see text] is a semilocal ring. We denote the [Formula: see text]-intersection graph of ideals of the ring [Formula: see text] by [Formula: see text], where [Formula: see text] are integers and [Formula: see text] is a [Formula: see text]-module. We determine the values of [Formula: see text] and [Formula: see text] for which [Formula: see text] is perfect. Furthermore, we derive a sufficient condition for [Formula: see text] to be weakly perfect.

Additive Maps on Units of Rings

Let R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Commutative rings with infinitely many maximal subrings

It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

EXTENSIONS OF HOMOGENEOUS SEMILOCAL RINGS I

A ring R with Jacobson radical J(R) is a homogeneous semilocal ring if R/J(R) is simple artinian. In this paper, we study the transfer of the property of being homogeneous semilocal from a ring R to the formal power series ring R[[x]], the skew formal power series ring R[[x, α]] and the Hurwitz series ring HR. The results of the paper generalize those proved for commutative local rings. We also consider finite centralizing extensions proving that if the ring of matrices Mn(R) is a homogeneous semilocal ring, then so is R. More generally, if e is an idempotent of a homogeneous semilocal ring S, then eSe is homogeneous semilocal.

HOMOGENEOUS SEMILOCAL GROUP RINGS AND CROSSED PRODUCTS

Let J(R) be the Jacobson radical of a ring R. Then R is called homogeneous semilocal if R/J(R) is simple artinian. The aim of this paper is to find necessary and sufficient conditions for the group rings and the crossed products to be homogeneous semilocal ring.

A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes II: Involution of the second kind

Let X be a regular noetherian scheme of finite Krull dimension with involution σ and an Azumaya algebra over X with involution τ of the second kind with respect to σ. We construct a hermitian and a skew-hermitian Gersten-Witt complex for (, τ) and show that these complexes are exact if X = Spec R is the spectrum of a regular semilocal ring R of geometric type, such that R is a quadratic étale extension of the fixed ring of σ.