Extensions of centrally essential rings

We show that all trivial (unital and essential) extensions of C (X) by a σ-unital purely infinite simple C*-algebra A with K1(A) = 0 are unitarily equivalent, provided that X is homeomorphic to a compact subset of the real line or the unit circle. Therefore all (unital and essential) extensions of such can be completely determined by Ext(B, A). An invariant is introduced to classify all such trivial (unital and essential) extensions of C (X) by a σ-unital C*-algebra A with the properties that RR (M (A)) = 0 and C (A) is simple.

All rings are assumed to be associative and (except for nil-rings and some stipulated cases) to have nonzero identity elements. Expressions such as a “Noetherian ring” mean that the corresponding right and left conditions hold. A module is said to be simple if it does not have nonzero proper submodules. A submodule N of a module M is called a maximal submodule (in M) if the factor module M/N is simple. A ring A is called a right max ring (or a right Bass ring) if every nonzero right A-module has a maximal submodule. A ring is said to be orthogonally finite if it does not contain an infinite set of nonzero orthogonal idempotents. Bass [11] proved that a ring with the minimum condition on principal left ideals is an orthogonally finite right max ring. Cozzens [25] and Koifman [61] constructed examples of orthogonally finite right max rings which do not satisfy the minimum condition on principal left ideals. This review contains some new and some old results related to max rings. We present the necessary notation and definitions. Let A be a ring. We denote the center of the ring A by C(A). A subset B of A is said to be central (in A) if B ⊆ C(A). If M and N are two modules, then Hom(M,N) denotes the Abelian group formed by all homomorphisms M → N . For a module M , we denote by End(M) the endomorphism ring Hom(M,M) of M . For a subset B of A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. An element a of A is said to be right regular (resp. left regular) in A if r(a) = 0 (resp. (a) = 0). A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is said to be right weakly regular if B2 = B for every right ideal B of A. A ring without nonzero nilpotent ideals is called a semiprime ring. A ring is called a prime ring if the product of any two of its nonzero ideals is not equal to zero. A ring A is said to be simple if each of its nonzero ideals coincides with A. A ring is called a domain if the product of any two of its nonzero elements is not equal to zero. For a ring A, an ideal P of A is called a prime (resp. semiprime and completely prime) ideal if the factor ring A/P is a prime ring (resp. a semiprime ring and a domain). A ring A is called a P.I.-ring (or a ring with polynomial identity) if A satisfies the polynomial identity f(x1, . . . , xn) = 0, where f(x1, . . . , xn) is a polynomial in noncommutative variables with coefficients in the ring of integers Z, and Z coincides with its ideal generated by the coefficients of f(x1, . . . , xn). A submodule N of a module M is said to be essential (in M) if N has nonzero intersection with any nonzero submodule of the module M . In this case, we say that M is an essential extension of the module N . A submodule N of a module M is called a superfluous submodule (in M) if N +M ′ = M for every proper submodule M ′ of M .

Comultiplication Modules over Noncommutative Rings

Let [Formula: see text] be a commutative ring with nonzero identity element and [Formula: see text] a multiplicative subset of [Formula: see text]. In this paper, we introduce and investigate the notion of nonnil [Formula: see text]-SFT rings. The ring [Formula: see text] is said to be nonnil [Formula: see text]-SFT, if for each nonnil-ideal [Formula: see text] of [Formula: see text], there exist [Formula: see text], [Formula: see text] and a finitely generated ideal [Formula: see text] such that [Formula: see text] for all [Formula: see text], in that case, the ideal [Formula: see text] is called [Formula: see text]-SFT. It is shown that the ring [Formula: see text] is nonnil [Formula: see text]-SFT ring if and only if each nonnil-prime ideal (disjoint with [Formula: see text]) is [Formula: see text]-SFT. The transfert of the nonnil [Formula: see text]-SFT concept to flat overrings and trivial extension is investigated. We give several characterizations of nonnil [Formula: see text]-SFT rings. Also, we give a characterization of the nonnil-SFT variant in terms of the nonnil [Formula: see text]-SFT variant.

A ring [Formula: see text] is said to be centrally essential if for every its nonzero element [Formula: see text], there exist nonzero central elements [Formula: see text] and [Formula: see text] with [Formula: see text]. A ring [Formula: see text] is said to be completely centrally essential if all its factor rings are centrally essential rings. It is proved that completely centrally essential semiprimary rings are Lie nilpotent; noetherian completely centrally essential rings are strongly Lie nilpotent (in particular, every such ring is a [Formula: see text]-ring). Every completely centrally essential ring has the classical ring of fractions which is a completely centrally essential ring. If [Formula: see text] is a commutative domain and [Formula: see text] is an arbitrary group, then any completely centrally essential group ring [Formula: see text] is commutative.

All rings are assumed to be associative and (except for nil-rings and for some stipulated cases) to have nonzero identity elements. A ring A is an exchange ring if the following two equivalent conditions hold: (1) for any element a ∈ A, there exists an idempotent e ∈ aA with 1 − e ∈ (1 − a)A; (2) for any element a ∈ A, there exists an idempotent e ∈ Aa with 1− e ∈ A(1− a). (The equivalence (1)⇐⇒(2) is proved in 2.1 below.) A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is semiregular if A/J(A) is a regular ring and all idempotents of A/J(A) can be lifted to idempotents of A. A ring A is a π-regular ring if, for every element a of A, there exists an element b of A such that a = aba for some positive integer n. Every semiregular or π-regular ring is an exchange ring (see 2.11(1)). It is directly verified that the ring of all rational numbers with odd denominators is an exchange ring that is not a π-regular ring. Let B be the ring of all rational numbers with odd denominators, {Ai}i=1 be a countable infinite set of copies of the field of rational numbers, D be the direct product of all rings Ai, and R be the subring in D generated by the ideal ⊕i=1Ai and by the subring {(b, b, b, . . . ) | b ∈ B}. Then R is a commutative reduced semiprimitive exchange ring that is not a semiregular ring (see 2.9(4)). A moduleM has the finite exchange property if for every moduleX and for every direct decomposition X = M ′ ⊕ Y = ⊕i∈INi, where M ′ ∼= M and I is a finite set, there exist submodules N ′ i ⊆ Ni (i ∈ I) such that X = M ′ ⊕ (⊕i∈IN ′ i). The importance of exchange rings is related to the fact that a module M has the finite exchange property if and only if the ring End(M) is an exchange ring [75]. The systematic study of modules with the finite exchange property and exchange rings was initiated in [18,58,75]. Also, see [2–4,14,17,39,43,48,55,56,58–61,63,67,73–75,77–79,82–85,87–90,93]. Exchange rings are potent rings (a ring A is potent if every right ideal of A that is not contained in J(A) contains a nonzero idempotent and all idempotents of A/J(A) can be lifted to idempotents of A). Thus, exchange rings have many idempotents. For a module M , the Jacobson radical, the endomorphism ring, and the lattice of all submodules are denoted by J(M), End(M), and Lat(M), respectively. For a ring A, C(A) and U(A) denote the center and the group of invertible elements of A, respectively. For a subset B of a ring A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. If X and Y are subsets of a ring A, then XY denotes the set of all finite sums { ∑ xiyi | xi ∈ X, yi ∈ Y }. In particular, ABA denotes the ideal of A generated by its subset B. A submoduleN of a moduleM is essential (in M) if N has the nonzero intersection with any nonzero submodule of M . In this case, we say that the module M is an essential extension of N . A right (resp. left) module M over a ring A is a nonsingular module if r(m) (resp. (m)) is not an essential right (resp. left) ideal of A for every nonzero element m of M . A ring without nonzero nilpotent elements is called a reduced ring. A ring is normal if all its idempotents are central. Every reduced ring A is normal, since for any idempotent e ∈ A, we have 0 = (eA(1 − e))2 = ((1− e)Ae)2, whence eA(1− e) = (1− e)Ae = 0.

Let q be a power of a prime. The lattice of one-sided ideals of the finite unital non-commutative Frobenius ring M2(ℱq) of 2 × 2 matrices over the Galois field (ℱq) is completely analyzed. It turns out that M2(ℱq) is a principal left semi-local ring in which each left ideal is generated by an idempotent element. The explicit forms of the non-trivial idempotents of M2(ℱq) are determined to give q + 1 proper non-trivial left maximal ideals each with q elements. These are exactly the minimal left ideals as well. Using the structure of M2(ℱq) as a partial ordering of ideals, the generalized Möbius and Euler phi functions are applied to derive the explicit form of the homogeneous weight function on M2(ℱq). This weight depends on whether the element is the zero element, a zero divisor or a unit. A zero divisor gives the largest homogeneous weight. Moreover, orbit codes over M2(ℱq) are constructed via the action of the general linear group GL(2, q) on M2(ℱq) by left translation. The orbit determined by a nonzero nonunit idempotent element of M2(ℱq) forms the nonzero elements of a minimal left ideal of M2(ℱq) which are all zero divisors. Consequently, it is shown that the minimum homogeneous distance of the orbit code generated by a nonzero nonunit idempotent element of M2(ℱq) approaches the Plotkin upper bound as the field size q becomes larger. Analogous results are obtained when the lattice of right ideals is considered and the action of GL(2, q) on M2(ℱq) by right translation is used instead.

Based on McCoy’s theorem on commutative rings, Nielsen called a ring R right McCoy if the equation f(x)g(x)=0 implies f(x)c=0 for some nonzero element c in R, where f(x) and g(x) are nonzero polynomials in R[x]. In this paper, a class of rings is introduced and called ZPZC rings, containing McCoy rings, and then their properties are investigated. Also, associations are found between ZPZC rings and other related rings. Moreover, several extensions of ZPZC rings are studied, including matrix rings, trivial extensions, Hochschild extensions and classical quotient rings.

A nonzero element a in a unital ring R is called unit fine if there is a unit u in R such that ua is fine (i.e. a sum of a unit and a nilpotent). A ring all whose elements are unit fine is called accordingly. This turns out to be a new class of simple rings, including the class of fine rings (and so the class of simple Artinian rings). The paper studies unit fine elements and rings. For rings such that 1 is a sum of two units, it is proved that matrix rings over unit fine rings, are unit fine.

Let be an absolute valued algebra containing a nonzero central algebraic element. Then is a pre-Hilbert algebra and is finite dimensional in the following cases: 1) A satisfies (x, x, x)=0. 2) A satisfies (x2, x2 , x2 )=0. 3) A satisfies (x, x2, x)=0. In these cases is isomorphic to or . It may be conjectured that every absolute valued algebra containing a nonzero central element is pre-Hilbert algebra.

We study remarkable sub-lattice effect algebras of Archimedean atomic lattice effect algebras E, namely their blocks M, centers C(E), compatibility centers B(E) and sets of all sharp elements S(E) of E. We show that in every such effect algebra E, every atomic block M and the set S(E) are bifull sub-lattice effect algebras of E. Consequently, if E is moreover sharply dominating then every atomic block M is again sharply dominating and the basic decompositions of elements (BDE of x) in E and in M coincide. Thus in the compatibility center B(E) of E, nonzero elements are dominated by central elements and their basic decompositions coincide with those in all atomic blocks and in E. Some further details which may be helpful under answers about the existence and properties of states are shown. Namely, we prove the existence of an (o)-continuous state on every sharply dominating Archimedean atomic lattice effect algebra E with $$B(E)\not =C(E).$$ Moreover, for compactly generated Archimedean lattice effect algebras the equivalence of (o)-continuity of states with their complete additivity is proved. Further, we prove “State smearing theorem” for these lattice effect algebras.

Let A be an amenable separable C*-algebra and B be a non-unital but σ-unital simple C*- algebra with continuous scale. We show that two essential extensions τ1 and τ2 of A by B are approximately unitarily equivalent if and only ifIf A is assumed to satisfy the Universal Coefficient Theorem, there is a bijection fromapproximate unitary equivalence classes of the abovementioned extensions to KL(A,M(B)/B). Using KL(A,M(B)/B), we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

Ideal extensions and directly infinite algebras

In this article, basic left (right) ideals of Leavitt Path Algebra over a commutative unital ring are studied. We give conditions under which a basic left (right) ideal generated by a vertex is a minimal basic left (right) ideal. It is further shown that if R has no non-zero nilpotent elements, then every minimal basic left ideal \(L_R(E)x\) of the Leavitt path algebra \(L_R(E)\) contains a vertex. Among other techniques, the proof depends on the fact that a Leavitt Path Algebra over a commutative unital ring R is non-degenerate if and only if R has no non-zero nilpotent elements (equivalently, R is a (commutative) semiprime ring).KeywordsLeavitt path algebraBasic left idealMinimal basic left ideal

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.