Commutative normed rings

Part One: The General Theory of Commutative Normed Rings: 1.1 The concept of a normed ring; 1.2 Maximal ideals; 1.3 Abstract analytic functions; 1.4 Functions on maximal ideals. The radical of a ring; 1.5 The space of maximal ideals; 1.6 Analytic functions of an element of a ring; 1.7 The ring $\hat{R}$ of functions $x(M)$; 1.8 Rings with an involution The General Theory of Commutative Normed Rings (cont'd): 2.9 The connection between algebraic and topological isomorphisms; 2.10 Generalized divisors of zero; 2.11 The boundary of the space of maximal ideals; 2.12 Extension of maximal ideals; 2.13 Locally analytic operations on certain elements of a ring; 2.14 Decomposition of a normed ring into a direct sum of ideals; 2.15 The normed space adjoint to a normed ring Part Two: The Ring of Absolutely Integrable Functions and Their Discrete Analogues: 3.16 The ring $V$ of absolutely integrable functions on the line; 3.17 Maximal ideals of the rings $V$ and $V_+$; 3.18 The ring of absolutely integrable functions with a weight; 3.19 Discrete analogues to the rings of absolutely integrable functions Harmonic Analysis on Commutative Locally Compact Groups: 4.20 The group ring of a commutative locally compact group; 4.21 Maximal ideals of the group ring and the characters of a group; 4.22 The uniqueness theorem for the Fourier transform and the abundance of the set of characters; 4.23 The group of characters; 4.24 The invariant integral on the group of characters; 4.25 Inversion formulas for the Fourier transform; 4.26 The Pontrjagin duality law; 4.27 Positive-definite functions The Ring of Functions of Bounded Variation on a Line: 5.28 Functions of bounded variation on a line; 5.29 The ring of jump functions; 5.30 Absolutely continuous and discrete maximal ideals of the ring $V^{(b)}$; 5.31 Singular maximal ideals of the ring $V^{(b)}$; 5.32 Perfect sets with linearly independent points. The asymmetry of the ring $V^{(b)}$; 5.33 The general form of maximal ideals of the ring $V^{(b)}$ Part Three: Regular Rings: 6.34 Definitions, examples, and simplest properties; 6.35 The local theorem; 6.36 Minimal ideals; 6.37 Primary ideals; 6.38 Locally isomorphic rings; 6.39 Connection between the residue-class rings of two rings of functions, one embedded in the other; 6.40 Wiener's Tauberian theorem; 6.41 Primary ideals in homogeneous rings of functions; 6.42 Remarks on arbitrary closed ideals. An example of L. Schwartz Rings with Uniform Convergence: 7.43 Symmetric subrings of $C(S)$ and compact extensions of a space $S$; 7.44 The problem of arbitrary closed subrings of the ring $C(S)$; 7.45 Ideals in rings with uniform convergence Normed Rings with an Involution and Their Representations: 8.46 Rings with an involution and their representations; 8.47 Positive functionals and their connection with representations of rings; 8.48 Embedding of a ring with an involution in a ring of operators; 8.49 Indecomposable functionals and irreducible representations; 8.50 The case of commutative rings; 8.51 Group rings; 8.52 Example of an unsymmetric group ring The Decomposition of a Commutative Normed Ring into a Direct Sum of Ideals: 9.53 Introduction; 9.54 Characterization of the space of maximal ideals of a commutative normed ring; 9.55 A problem on analytic functions in a finitely generated ring; 9.56 Construction of a special finitely generated subring; 9.57 Proof of the theorem on the decomposition of a ring into a direct sum of ideals; 9.58 Some corollaries Historico-Bibliographical Notes Bibliography Index.