This work presents advances in zero-dimensional commutative rings and commutative algebra. It illustrates the research frontier with 52 open problems together with comments on the relevant literature, and offers a comprehensive index for easy access to information. Wide-ranging developments in commutative ring theory are examined.
Background and preliminaries on zero-dimensional rings; zero-dimensionality and products of commutative rings; zero-dimensional extension rings and subrings; residue fields of zero-dimensional rings; dimensions of extension rings; Picard groups, cancellation and the multiplicative structure of fields; derivations and deformations of zero-dimensional rings and algebras; the ring R[X,r/X]; P.M. Cohn's completely primal elements; some factorization properties of the ring of integer-valued polynomials; on the nonuniversality of weak normalization; a visit to valuation and pseudo-valuation domains; answer to a question on the principal idea theorem; on the Davenport constant, the cross number, and their application in factorization theory; the Picard group of a polynomial ring; integral extensions with fibers of prescribed cardinality; homological dimensions of localizations of polynomial rings; splitting rings for p-local torsion-free groups; prime ideals in birational extensions of polynomial rings II; duals of ideals in pullback constructions; cancellation and prime spectra; the ascending chain condition on n-generated submodules; more almost Dedekind domains and Prufer domains of polynomials; direct limits of finite products of fields; Eisenstein-type irreducibility criteria; on integer-valued polynomials; the prime spectrum of an infinite product of zero dimensional rings; when is a self-injective ring zero-dimensional?; some problems in commutative ring theory.