Barbara L. Osofsky

Cyclic modules whose quotients have all complement submodules direct summands

In [20,21], Osofsky showed that a ring all of whose cyclic modules are injective is semisimple Artinian. Since that time, a cyclic, finitely presented module version of the theorem has been proved in [S] and used to classify certain kinds of rings (see [ 8,9]). Also, Ahsan [ 1 ] applied the proof in [21] to rings all of whose cyclic modules are quasi-injective, enabling Koehler [ 171 to show that any ring with all cyclic modules quasi-injective is a product of a semisimple Artinian ring and a noncommutative analog of the injective pre-self-injective rings of Klatt and Levy [16]. Goel and Jain [ 1 l] applied the proof of [21] in their study of nonsingular and selfinjective rings with every cyclic quasi-continuous. Also, several authors have studied rings for which hypotheses on cyclic modules show that cyclic singular modules are injective (see [9, 10, 24, 251). In this paper we eliminate extraneous hypotheses used in previous proofs of the theorem in [20,21] to get a very general result. We prove that a cyclic module A4 has finite uniform dimension if all quotients of cyclic submodules of M have the property that all complement submodules are direct summands. The modules studied need not have endomorphism rings which are von Neumann regular (at least modulo their Jacobson radicals), a property

Upper bounds on homological dimensions

The homological dimension of a module M R is often related to the cardinality of a set of generators for M or for right ideals of R . In this note, upper bounds for this homological dimension are obtained in two situations.

On twisted polynomial rings

By suitably modifying the field of a twisted polynomial ring, we show that P, and Pa are completely independent, and that P, and Pa may simultaneously hold. We also show that Pa may hold on the right but not on the left. The following meanings of p, F, R, R, and D will be assumed throughout the entire paper. Let F be a field of characteristic p > 0, and let cr : F ---f F be the endomorphism of F defined by cr(a) = G for all 01 c F. Let i? be the ring of twisted polynomials with coefficients on the left,

A semiperfect one-sided injective ring

We construct an example of a ring R: such that i) R is semiperfect, ii) R is right but not left self-injective, iii) R is an essential extension of its socle on the left but not on the right.

Injective modules over twisted polynomial rings

Differential polynomial rings over a universal field and localized twisted polynomial rings over a separably closed field of non-zero characteristic twisted by the Frobenius endomorphism were the first domains not divisions rings that were shown to have every simple module injective (see [C] and [C-J]). By modifying the separably closed condition for the polynomial rings twisted by the Frobenius, the conditions of every simple being injective and only a single isomorphism class of simple modules were shown to be independent (see [O]). In this paper we continue the investigation of injective cyclic modules over twisted polynomial rings with coefficients in a commutative field.

Noether Lasker Primary Decomposition Revisited

There are many theorems which are absolutely basic for the study of particular areas of mathematics. Some of these theorems work their way into the standard undergraduate curriculum, but many are reserved for specialized graduate work. In this note, we present a theorem usually reserved for a first graduate course in commutative algebra in a way that undergraduates studying abstract algebra might find congenial. The theorem is the Noether-Lasker Primary Decomposition Theorem, which may be thought of as a way to extend a version of prime factorization from the integers to a much larger class of rings, including polynomials in several variables over a field. Traditionally, the approach taken to this theorem has been arithmetical, i.e. using the multiplicative properties of the ring. Although we refer to the arithmetic of the integers, our approach is more in the spirit of vector spaces. There are a large number of exercises to help the reader get involved in understanding the material. It is not cheating to get help on these exercises. We assume that the reader has had a first abstract algebra course, including the basic homomorphism theorems for groups and rings, some study of prime factorization for a Euclidean domain such as the integers or polynomials over a field, and at least the definition of a prime ideal in a commutative ring. The other necessary concepts from commutative algebra will be given here. The reader should also have seen the definition of vector space over a field and worked with bases and linear independence in a vector space. That is the background we build on. Although we prove a generalized version of the primary decomposition for finitely generated modules over a commutative Noetherian ring, including uniqueness of primary components if and only if the prime is a minimal associated prime, the reader does not have to know what those words mean to understand our main theorem. The major idea in the proof is contained in a useful module-theoretic definition. After we prove the theorem, we define all the concepts in the NoetherLasker Theorem so we can specialize our theorem to the case of interest in algebraic geometry and commutative algebra. A (left) module over a ring R with identity 1 is a system N between two R-modules satisfies the axioms for a linear transformation. To make the notation look more like the associative law rather than the commutative law, we will write our functions to the right of their arguments, on the side opposite the scalars. Although it does not

PROPERTIES OF INJECTIVE HULLS OF A RING HAVING A COMPATIBLE RING STRUCTURE

If the injective hull E = E ( R R ) of a ring R is a rational extension of R R , then E has a unique structure as a ring whose multiplication is compatible with R -module multiplication. We give some known examples where such a compatible ring structure exists when E is a not a rational extension of R R , and other examples where such a compatible ring structure on E cannot exist. With insights gleaned from these examples, we study compatible ring structures on E , especially in the case when E R , and hence R R ⊆ E R , has finite length. We show that for R R and E R of finite length, if E R has a ring structure compatible with R -module multiplication, then E is a quasi-Frobenius ring under that ring structure and any two compatible ring structures on E have left regular representations conjugate in Λ = End R ( E R ), so the ring structure is unique up to isomorphism. We also show that if E R is of finite length, then E R has a ring structure compatible with its R -module structure and this ring structure is unique as a set of left multiplications if and only if E R is a rational extension of R R .

Projective dimension is a lattice invariant

Abstract We show that, for a free abelian group G and prime power p ν , every direct sum decomposition of the group G / p ν G lifts to a direct sum decomposition of G . This is the key result we use to show that, for R a commutative von Neumann regular ring, and E a set of idempotents in R , then the projective dimension of the ideal E R as an R -module the same as the projective dimension of the ideal EB as a B -module, where B is the boolean algebra generated by E ∪{1} . This answers a 30 year old open question of R. Wiegand.

Nice Polynomials for Introductory Galois Theory

Dick and Jane are students in an advanced undergraduate modern algebra course. They are studying Galois theory. To help them we write down a family of polynomials {ff(x): n > 1) where f,(x) has degree i. We then show them how to check, using standard methods that they have used before, that the polynomials f (x) for primes p have Galois group the symmetric group on p letters. Lets see how. Dick and Jane are fortunate to have had a good high school background in mathematics. They have learned about prime factorization of integers since grade school. They know that a polynomial f(x) = Enc xi of degree n with rational coefficients has a rational root a if and only if the polynomial (x a) is a factor of p(x); that is, f(a) = 0 -=*f(x) = (x a)g(x) for some polynomial g(x) with rational coefficients. They also know that if f(x) is a nonconstant polynomial with integer coefficients, then any rational root P of f(x), where p and q are relatively prime

Projective Dimensions of Ideals of Prufer Domains

A Prufer domain has all of its finitely generated ideals projective. Countably but not finitely generated ideals must be of projective dimension 1. A standard argument due to Auslander gives an upper bound for the projective dimension of an ideal I. If I is Nk-generated, then I has projective dimension ≆ = k+1. Here we look at how one might get the reverse inequality.