Nonzero Socle Research Articles

On regular rings and V-rings

In this note, certain generalisations of strongly regular rings are considered in connection with regular rings andV-rings. The result that strongly regular rings are left (and right)V-rings [11] is extended. A condition for prime leftV-rings to be primitive with non-zero socle is given (this is related to a question ofFisher [7, Problem 3]. IfA is an ALD (almost left duo) ring, then (1) a simple leftA-module is injective iff it isp-injective; (2)A is von Neumann regular iff every maximal essential right ideal ofA isf-injective. Characterisations of semi-simple Artinian and simple Artinian rings are given in terms of regular andV-rings.

Trivial Extension of a Ring with Balanced Condition

A ring R is called QF-1 if every faithful R-module is balanced. In this paper we study commutative QF-1 rings. It is shown that a commutative QF-1 ring is local if and only if it is uniform. It is well known that commutative artinian QF-1 rings are QF, but Osofsky has constructed a nonartinian nonnoetherian commutative injective cogenerator, so QF-1, ring which is a trivial extension of a valuation ring. It is shown that if a trivial extension of a valuation ring is QF-1, then it has a nonzero socle. Furthermore such rings become injective cogenerator rings under certain conditions.

Jordan Rings with Nonzero Socle

Let $\mathcal {J}$ be a nondegenerate Jordan algebra over a commutative associative ring $\Phi$ containing $\tfrac {1}{2}$. Defining the socle $\mathcal {G}$ of $\mathcal {J}$ to be the sum of all minimal inner ideals of $\mathcal {J}$, we prove that $\mathcal {G}$ is the direct sum of simple ideals of $\mathcal {J}$. Our main result is that if $\mathcal {J}$ is prime with nonzero socle, then either (i) $\mathcal {J}$ is simple unital and satisfies DCC on principal inner ideals, (ii) $\mathcal {J}$ is isomorphic to a Jordan subalgebra $\mathcal {J}’$ of the plus algebra ${A^ + }$ of a primitive associative algebra A with nonzero socle S, and $\mathcal {J}’$ contains ${S^ + }$, or (iii) $\mathcal {J}$ is isomorphic to a Jordan subalgebra $\mathcal {J}''$ of the Jordan algebra of all symmetric elements H of a. primitive associative algebra A with nonzero socle S, and $\mathcal {J}''$ contains $H \cap S$. Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if $\mathcal {J}$ is simple then $\mathcal {J}$ contains a completely primitive idempotent if and only if either $\mathcal {J}$ is unital and satisfies DCC on principal inner ideals or $\mathcal {J}$ is isomorphic to the Jordan algebra of symmetric elements of a $*$-simple associative algebra A with involution $*$ containing a minimal one-sided ideal.

Jordan rings with nonzero socle

Let J \mathcal {J} be a nondegenerate Jordan algebra over a commutative associative ring Φ \Phi containing 1 2 \tfrac {1}{2} . Defining the socle G \mathcal {G} of J \mathcal {J} to be the sum of all minimal inner ideals of J \mathcal {J} , we prove that G \mathcal {G} is the direct sum of simple ideals of J \mathcal {J} . Our main result is that if J \mathcal {J} is prime with nonzero socle, then either (i) J \mathcal {J} is simple unital and satisfies DCC on principal inner ideals, (ii) J \mathcal {J} is isomorphic to a Jordan subalgebra J ′ \mathcal {J}’ of the plus algebra A + {A^ + } of a primitive associative algebra A with nonzero socle S, and J ′ \mathcal {J}’ contains S + {S^ + } , or (iii) J \mathcal {J} is isomorphic to a Jordan subalgebra J \mathcal {J} of the Jordan algebra of all symmetric elements H of a. primitive associative algebra A with nonzero socle S, and J \mathcal {J} contains H ∩ S H\, \cap \,S . Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if J \mathcal {J} is simple then J \mathcal {J} contains a completely primitive idempotent if and only if either J \mathcal {J} is unital and satisfies DCC on principal inner ideals or J \mathcal {J} is isomorphic to the Jordan algebra of symmetric elements of a ∗ * -simple associative algebra A with involution ∗ * containing a minimal one-sided ideal.

Trivial extension of a ring with balanced condition

A ring R is called QF-1 if every faithful R-module is balanced. In this paper we study commutative QF-1 rings. It is shown that a commutative QF-1 ring is local if and only if it is uniform. It is well known that commutative artinian QF-1 rings are QF, but Osofsky has constructed a nonartinian nonnoetherian commutative injective cogenerator, so QF-1, ring which is a trivial extension of a valuation ring. It is shown that if a trivial extension of a valuation ring is QF-1, then it has a nonzero socle. Furthermore such rings become injective cogenerator rings under certain conditions.

Group Rings with Non-Zero Socle

The aim of this thesis is to investigate the circumstances under which group rings over fields have non-zero socle, i.e. contain minimal one-sided ideals. After an introductory chapter, we consider the special case o₊ a periodic abelian group and a non-modular field (that is, a field of characteristic prime to the orders of the elements of the group). This special case, and the background material contained in Chapter III, serve as preparation for our principal results, which concern, locally finite groups. We establish necessary and sufficient conditions on an arbitrary field K and a locally finite group 0 for the group ring KG to contain minimal one-sided ideals: the most important condition is that G should be a Cernikov group. We then examine the structure of KG when these conditions are satisfied. Vie show that KG has a finite series of ideal3 each factor of which i3 a direct sum of quasi-Frobenius rings, and characterize the socle of KG. Me also classify indecomposable KG-modules, and determine (for countable but not necessarily locally finite groups G) necessary and sufficient conditions for all indecomposable KG-modules to be irreducible. In the final chapter we consider non-locally-finite groups, conjecturing that group rings of such groups never contain minimal one-sided ideals. We establish the truth of this conjecture for several classes of groups, and also consider semiartinian group rings.

Primitive rings with involution and pivotal monomials

With the aid of ultrafilters an example is constructed of a primitive ring with involution and zero socle in which the symmetric elementsS satisfys2c=s,c depending on and commuting withs (thusS satisfies the pivotal monomialx). However, by suitably restricting the definition of generalized pivotal monomial (essentially by making the variables linear) it is shown that a primitive ring with involution has nonzero socle if and only if the symmetric elements satisfy a restricted GPM.

𝐼-rings

A ring R R , possibly with no identity, is called an I 0 {I_0} -ring if each one-sided ideal not contained in the Jacobson radical J ( R ) J(R) contains a nonzero idempotent. If, in addition, idempotents can be lifted modulo J ( R ) , R J(R),R is called an I I -ring. A survey of when these properties are inherited by related rings is given. Maximal idempotents are examined and conditions when I 0 {I_0} -rings have an identity are given. It is shown that, in an I 0 {I_0} -ring R R , primitive idempotents are local and primitive idempotents in R / J ( R ) R/J(R) can always be lifted. This yields some characterizations of I 0 {I_0} -rings R R such that R / J ( R ) R/J(R) is primitive with nonzero socle. A ring R R (possibly with no identity) is called semiperfect if R / J ( R ) R/J(R) is semisimple artinian and idempotents can be lifted modulo J ( R ) J(R) . These rings are characterized in several new ways: among them as I 0 {I_0} -rings with no infinite orthogonal family of idempotents, and as I 0 {I_0} -rings R R with R / J ( R ) R/J(R) semisimple artinian. Several other properties are derived. The connection between I 0 {I_0} -rings and the notion of a regular module is explored. The rings R R which have a regular module M M such that J ( R ) = ann ⁡ ( M ) J(R) = \operatorname {ann} (M) are studied. In particular they are I 0 {I_0} -rings. In addition, it is shown that, over an I 0 {I_0} -ring, the endomorphism ring of a regular module is an I 0 {I_0} -ring with zero radical.

Torsion-free covers

This paper studies the existence and properties of a torsion-free cover with respect to a faithful hereditary torsion theory (T, F) of modules over a ring with unity. A direct sum of a finite number of torsion-free covers of modules is the torsion-free cover of the direct sum of the modules. The concept of aT-near homomorphism, which generalizes Enochs’ definition of a neat submodule, is introduced and studied. This allows the generalization of a result of Enochs on liftings of homomorphisms. Hereditary torsion theories for which every module has a torsion-free cover are called universally covering. If the inclusion map ofR into the appropriate quotient ringQ is a left localization in the sense of Silver, the problem of the existence of universally-covering torsion theories can be reduced to the caseR=Q. As a consequence, many sufficient conditions for a hereditary torsion theory to be universally covering are obtained. For a universally-covering hereditary torsion theory (T, F), the following conditions are equivalent: (1) the product ofF-neat homomorphisms is alwaysT-neat; (2) the product of torsion-free covers is alwaysT-neat; (3) every nonzero module inT has a nonzero socle.

Rings Having Solvable Adjoint Groups

Let $^\circ R$ denote the group of quasi-regular elements of a ring $R$ with respect to circle operation. The following results have been proved: (1) If $R$ is a perfect ring and $^\circ R$ is finitely generated solvable group then $R$ is finite and hence $^\circ R = {P_1} \circ {P_2} \circ \; \cdots \; \circ {P_m}$ where ${P_i}$ are pairwise commuting $p$-groups. (2) Let $R$ be a locally matrix ring or a prime ring with nonzero socle. Then $\circ R$ is solvable iff $R$ is either a field or a $2 \times 2$ matrix ring over a field having at most $3$ elements.

Rings having solvable adjoint groups

Let $^\circ R$ denote the group of quasi-regular elements of a ring $R$ with respect to circle operation. The following results have been proved: (1) If $R$ is a perfect ring and $^\circ R$ is finitely generated solvable group then $R$ is finite and hence $^\circ R = {P_1} \circ {P_2} \circ \; \cdots \; \circ {P_m}$ where ${P_i}$ are pairwise commuting $p$-groups. (2) Let $R$ be a locally matrix ring or a prime ring with nonzero socle. Then $\circ R$ is solvable iff $R$ is either a field or a $2 \times 2$ matrix ring over a field having at most $3$ elements.

On commutative nonassociative algebras

In this paper we study a class of commutative nonassociative algebras which includes those special Jordan algebras which arise as the set of all elements fixed by an involution in a primitive ring with nonzero socle and with centralizer which is a field of characteristic not 2. Perhaps the most unusual feature of our approach is that we do not assume that our algebras satisfy any identity, but only that enough primitive idempotents exist satisfying certain properties that follow from the Jordan identity. (For an introduction to the standard theory, see [6].) We also need an axiom which insures that the discrete topology (Jacobson’s finite topology) will suffice. More specifically, we assume that the following four axioms on idempotents hold in our algebra A over the field @ (A will be assumed to be commutative and of characteristic not 2 hereafter):