Inducing lattice maps by semilinear isomorphisms

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.

Structural Properties of Models of N1-Categorical Theories

Ring Homomorphisms and Isomorphisms

Chapter 2 - Projective and Affine Geometry over Division Rings

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

This thesis studies the conditions under which certain U (G)-modules have integral dimension. Specifically, the paper explores the circumstances under which certain U (G)-modules are free modules. Two main theorems are proven for this endeavor. The first theorem takes the matrix representation A of a submodule N of a free left module M over an arbitrary ring R. The theorem states that if A can be put into reduced row echelon form, then N and M/N are both free left R-modules. The second theorem takes the matrix representation B of an arbitrary left R-module homomorphism T. If B can be put into reduced column echelon form, then Ker( T) is a free left R-module as well. The results of these theorems are well-known if R is a field or division ring. The results are also probably known for an arbitrary ring R , but there does not seem to be a readily accessible source in the standard literature.

Consider a left module V over a possibly noncommutative ring JR. The objective is to investigate finite or infinite sequences of submodules of V of the form {0} = Λo Q Ax Q A2 or of the form V = A° 2 A1 2 Az where all the quotient modules Ai+JAi or AiIAi+1 are completely reducible. It is shown that some of the known properties of such series for a module over a ring with minimum condition hold for a more general class of rings, a class which properly includes those satisfying the descending chain condition. The main difficulty which this note has attempted to solve is to generalize these well known theorems from the minimum condition case to a much larger class of rings and modules. The class of rings considered in this note seems to be the natural setting in which to prove these theorems. In spite of the added generality, our proofs are not longer than they would be if the minimum condition were assumed. All modules considered here will be understood to be left modules. A module V over a ring will be called simple provided it contains no proper nonzero submodules and provided also RV — Y Φ {0}. A module is completely reducible provided it is a finite or infinite direct algebraic

The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let is a commutative ring and is a divided prime ideal of Let be a ring with total quotient ring T(R), and set such that for every and every Then is a ring homomorphism from T(R) into and restricted to R is also a ring homomorphism from R into given by for every At this setting, we introduce the notions of -piecewise Noetherian rings and -piecewise w-Noetherian rings and we show that the theories of -piecewise Noetherian (resp. -piecewise w-Noetherian) rings resemble those of piecewise Noetherian (resp. piecewise w-Noetherian) domains.

Transfer of homological properties under base change is a classical field of study. Let <TEX>$R{\rightarrow}S$</TEX> be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between <TEX>$U{\otimes}^L_RX$</TEX>(or <TEX>$RHom_R(X,U)$</TEX>) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities <TEX>$G-dim_S(U{\otimes}^L_RX)=G-dim_RX+pd_SU$</TEX> and <TEX>$Gid_S(RHom_R(X,U))=G-dim_RX+id_SU$</TEX> hold.

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or else a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Lattice isomorphisms of finite one-generated rings with identity are studied. We elucidate the general structure of finite one-generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one-generated ring. We look at lattice isomorphisms of one-generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form GR(pn,m), where m > 1 and n > 1. We specify sufficient conditions for the projective image of a one-generated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.

A covariant functor has been defined in [1] which assigns to a commutative ring R an abelian monoid m(R). The elements of m(R) are equivalence classes (called homotopy classes) of homomorphisms of progenerator modules over R. The monoid homomorphism m(σ): m(R) → m(S) associated with a homomor-phism of rings σ: R → S is studied. Both necessary condi­tions and sufficient conditions for m(σ)1 {1} = {1}, for m(σ) to be injective and for m(σ) to be surjective are presented.

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

Free left modules over right Bezout domains are characterized by a set of geometric axioms resembling those of U. Brehm, where results of this kind have been obtained for torsion-free left modules over left Ore domains. Discourses by S. E. Schmidt and the present author have come to similar results for other classes of rings and modules.

Solution to a conjecture by Hofmeier–Wittstock

Let K be a field. We obtain conditions for elements of the form {1 + a i, 1 + b j} to generate a free group of rank 2 in a quaternion algebra over K, including the case where the characteristic of the field K is 2, based on an article by Professors Jairo Gonçalves, Arnaldo Mandel and Mazi Shirvani. These results are then applied to obtain pairs of elements that freely generate a free group in many other classes of division rings, such as total fields of fractions of Ore domains and Malcev-Neumann series rings. With these results, we partially answer a conjecture by A. Lichtman, regarding the existence of non-abelian free groups in the multiplicative group of non-commutative division rings.