On w_(∞)-Warfield Cotorsion Modules and Krull Domains

Let R be a commutative domain with 1 and Q(≠R)its field of quotients.In this note an R-module M is called w_(∞)-Warfield cotorsion if M∈WC∩P^(⊥)_(w_(∞)),where WC denotes the class of all Warfield cotorsion R-modules and P_(w_(∞))the class of all w_(∞)-projective R-modules.It is shown that R is a PVMD if and only if all w-cotorsion R-modules are w_(∞)-Warfield cotorsion,and that R is a Krull domain if and only if every w-Matlis cotorsion strong w-module over R is a w_(∞)-Warfield cotorsion w-module.

Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let K = Q（√d）, where Q is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over Q. For several quadratic fields K = Q（√d）, the ring Rd of integers of K is not a unique-factorization domain. For d 〈 0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = -1,-2,-3,-7,-11,-19,-43,-67,-163. Let Q denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/（Q^n） was determined by Cross in 1983 for the case d = -1. This paper completely determined the unit groups of Rd/（Q^n） for the cases d = -2, -3.

An Upper Bound for the w-Weak Global Dimension of Pullbacks

Let R be a commutative ring with identity and I0 an ideal of R.We introduce and study the c-weak global dimension c-w.gl.dim(R/I0)of the factor ring R/I0.Let T be a w-linked extension of R,and we also introduce the wR-weak global dimension wR-w.gl.dim(T)of T.We show that the ring T with wR-w.gl.dim(T)=0 is exactly a field and the ring T with wR-w.gl.dim(T)≤1 is exactly a PwRMD.As an application,we give an upper bound for the w-weak global dimension of a Cartesian square(RDTF,M).More precisely,if T is w-linked over R,then w-w.gl.dim(R)≤max{wR-w.gl.dim(T)+w-fdR T,c-w.gl.dim(D)+w-fdn D}.Furthermore,for a Milnor square(RDTF,M),we obtain w-w.gl.dim(R)≤max{wR-w.gl.dim(T)+w-fdR T,w-w.gl.dim(D)+w-fdR D}.

Notes on Rings with Strong 2-Sum Property

A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units.In this note,we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property.