A Boolean-like ring R is a commutative ring with unity in which 2x = 0 and xy(1 + x)(1 + y) = 0 hold for all elements x, y of the ring R. It is shown in this paper that in the category of Boolean-like rings, R is injective if and only if R is a complete Boolean ring and R is projective if and only if R = {0, 1}.

The construction, for a module M over a commutative ring A (with identity) and a multiplicatively closed subset S of A, of the module of fractions S-1M is, of course, one of the most basic ideas in commutative algebra. The purpose of this note is to present a generalization which constructs, for a positive integer n and what is called a triangular subset U of An = A × A × … × A (n factors), a module U-n M of generalized fractions, a typical element of which has the form

where m∈ M and (u1, … un)∈U.

A module M over a ring R is κ-projective, κ a cardinal, if M is projective relative to all exact sequence of R-modules 0 → A → B → C → 0 such that C has a generating set of cardinality less than κ. A structure theorem for κ-projective modules over Dedekind domains is proven, and the κ-projectivity of M is related to properties of ExtR (M, ⊕ R). Using results of S. Chase, S. Shelah and P. Eklof, the existence of non-projective и1-projective modules is shown to undecidable, while both the Continuum Hypothesis and its denial (Plus Martin's Axiom) imply the existence of a reduced И0-projective Z-module which is not free.

Let R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

Let N be a direct summand of a module which is a direct sum of modules of torsion-free rank one over a discrete valuation ring. Then there is a torsion module T such that N⊕T is also a direct sum of modules of torsion-free rank one.

Let L/K be a quadratic extension of algebraic number fields, and D a central L-division algebra of finite L-dimension d2. If - is an involution (i.e., a ring antiautomorphism of period two) of D, we write S(-) for the set of - symmetric elements of D:

Let R be a commutative ring with identity, and let U be a unitary commutative R-algebra with identity. In [1] Gilmer defines the (l/n)th power (n a positive integer) of a valuation ideal R when R is a domain. Sections 2, 3 of the present note are devoted to the study of an extension of this notion to positive rational powers of an arbitrary R-submodule of U.