… present paper, we mean by a ring a commutative ring with identity and by a module a unitary
one. Let R be a ring and let A be a homomorphic image of the polynomial ring R [X ] of a set …

… In this note we seek some conditions which are necessary and sufficient in order that S is,
as an R-module, faithfully flat. They will be characterized in terms of the R-module I , which is …

… In this vein, we show in Corollary 2.5 that for each positive integer n, flatness is … 2-flatness
is not a universal property. Throughout, all rings and algebras are assumed commutative, with …

… reduced, von Neumann regular PI ring and the order of G … commutative von Neumann
regular; R is reduced von Neumann regular and G is solvable (Jondrup [8]).• The skew group ring …

… Let I denote an ideal of a commutative ring A. For our purposes we need the local cohomology
functors Hj( ), i E 1, of A with respect to I. See Grothendieck [5] for basic facts about them. …

… We give necessary and sufficient conditions for the projectivity and flatness over the graded
ring of semi-coinvariants of A. When A and C are commutative and G is any subgroup of the …

… We note that the ring A is reduced. Indeed, if a polynomial f (yl,. . . , y,) in B maps to a
nilpotent element in A we must have that f (61,. . . ,… We have a commutative diagram …

… the trivial extension of ℤ 2 by A is a commutative ring with Soc(R) = 0∝ A. Now, let M =
Soc(R) = 0∝ A. It follows that every left R-module is M-flat since M is semisimple and hence …

… for the projectivity and flatness over the graded ring of normalizing elements of A. When A
is not necessarily commutative we obtain similar results over the graded ring of weakly semi-…

… the interaction between flatness and adic completion for infinitely generated modules over
a commutative ring, that we had started in [19]. Let A be a commutative ring, and let a be an …