On projective modules over polynomial rings

Journal of Pure and Applied Algebra, 2003

Let A be a Noetherian ring of Krull dimension n containing the ÿeld of rationals. Let P be a projective A[T ]-module of rank n with trivial determinant such that the A-module P=TP has a free summand of rank one. It is proved that if n is even, then P has a free summand of rank one if it maps onto an ideal I of A[T ] of height n which is generated by n elements.

Journal of Algebra, 1971

2000

UDC 512.553

Sbornik: Mathematics, 1998

Let A be a hereditary Noetherian prime ring that is not right primitive. A complete description of π-injective A-modules is obtained. Conditions under which the classical ring of quotients of A is a π-projective A-module are determined. A criterion for a right hereditary right Noetherian prime ring to be serial is obtained. Bibliography: 8 titles. All rings in this paper are assumed to be associative and with a non-zero identity element. Expressions such as 'a Noetherian ring' mean that the corresponding right-hand and left-hand conditions hold. The Jacobson radical of a module M is denoted by J(M). A module is hereditary if all its submodules are projective. A module is uniserial if any two of its submodules are comparable with respect to inclusion. A module M is serial if M is a direct sum of uniserial modules. A uniserial Noetherian domain A is complete if the ring A is complete with respect to the J(A)-adic topology. A module M is projective with respect to a module N (or N-projective) if for each epimorphism h: N → N and each homomorphism f : M → N there exists a homomorphism f : M → N such that hf = f. A module that is projective with respect to itself is called a quasiprojective (or self-projective) module. A module M is π-projective (see [1], p. 359) if for arbitrary submodules U and V of M such that U + V = M there exists an endomorphism f of M such that f(M) ⊆ U and (1 − f)(M) ⊆ V. If all factor modules of a module M are indecomposable, then M is clearly π-projective. Hence all uniserial modules are π-projective. Each quasiprojective module is π-projective [1], p. 359. Each quasicyclic Abelian group C(p ∞) is a π-projective non-quasiprojective module over the ring of integers. Recall that each Noetherian prime ring A has a simple Artinian classical ring of quotients Q, and Q A is an injective non-singular right (left) A-module. Conditions ensuring that the classical ring of quotients of a hereditary Noetherian prime, but not right primitive ring A is a quasiprojective right A-module are described in [2], Theorem 8. In this connection we present the following Theorems 1 and 2, which are the first two main results of this paper.

2008

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3 investigates the correlation with well-known Prüfer conditions; namely, we prove that this class of rings stands strictly between the two classes of arithmetical rings and Gaussian rings. Thereby, we generalize Osofsky's theorem on the weak global dimension of arithmetical rings and partially resolve Bazzoni-Glaz's related conjecture on Gaussian rings. We also establish an analogue of Bazzoni-Glaz results on the transfer of Prüfer conditions between a ring and its total ring of quotients. Section 4 examines various contexts of trivial ring extensions in order to build new and original examples of rings where all finitely generated ideals are subject to quasi-projectivity, marking their distinction from related classes of Prüfer rings.

2010

Let $A$ be a ring of dimension $d$. Assume that for every finite extension ring $R$ of $A$, E_{d+1}(R) acts transitively on Um_{d+1}(R). Then we prove that E(A\oplus P) acts transitively on Um(A\oplus P), for any projective A-module P of rank d. As a consequence of this, we generalise some results of Gubeladze.

2010

In the case of Laurent polynomial ring (i.e. f_i=Y_i), (i) is due to Lindel, (ii) is due to Bhatwadekar, Lindel and Rao and (iii, iv) is due to Suslin.

Proceedings of the American Mathematical Society, 1993

Transactions of the American Mathematical Society, 2013

We use pullbacks of rings to realize the submonoids M of (N0 ∪ {∞}) k which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right R-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left R-modules is determined by the monoid D(M ) defined by reversing the inequalities determining the monoid M . These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters . We also provide a rich variety of examples of semilocal rings having non finitely generated projective modules that are finitely generated modulo the Jacobson radical.

International Journal of Open Problems in Computer Science and Mathematics, 2012

A ring extension R ⊂ S is said to satisfy the (FMC) condition if there is a finite maximal chain of rings from R to S (cf. [1]). The aim of this note is to prove that if R ⊂ S is an extension of integral domains satisfying the (FMC) condition, then S is a P-extension of R. As a consequence, we establish that the polynomial extension R[X] ⊂ S[X] does never satisfy the (FMC) condition.