Essential extension

Abstract: The notion of a direct summand of a ring containing the set of nilpotents in some "dense" way has been considered by Y. Utumi, L. Jeremy, C. Faith, and G. F, Birkenmeier. Several types of rings including right selfinjective rings, commutative FPF rings, and rings which are a direct sum of indecomposable right ideals have been shown to have a MDSN (i.e., the minimal direct summand containing the nilpotent elements). In this paper, the class of rings which have a MDSN is enlarged to include quasiBaer rings and right quasi-continuous rings. Also, several known results are generalized. Specifically, the following results are proved: (Theorem 3) Let R be a ring in which each right annihilator of a reduced (i.e., no nonzero nilpotent elements) right ideal is essential in an idempotent generated right ideal. Then R^A®B where B is the MDSN and an essential extension of Nt (i.e., the ideal generated by the nilpotent elements of index two), and A is a reduced right ideal of R which is also an abelian Baer ring. (Corollary 6) Let R be an ATF*-aIgebra. Then R = A®B where A is a commutative AW*-algebra, and B is the MDSN of R and B is an ATF*-algebra which is a rational extension of Nt. Furthermore, A contains all reduced ideals of R. (Theorem 12) Let R be a ring such that each reduced right ideal is essential in an idempotent generated right ideal. Then R = A 0 B where B is the densely nil MDSN, and A is both a reduced quasi-continuous right ideal of R and a right quasi-continuous abelian Baer ring.

Abstract: 0. Introduction and preliminaries. Following R. E. Johnson [1] we assume in this paper that any rings we shall be concerned with satisfy either one or both of the following conditions: (Jf) If the right annihilator of a left ideal A is nonzero, then there exists a nonzero left ideal B such that A nB =0. (J) =the right left symmetry of (J1).I We say that a ring is a Ji-ring, a Jf-ring or a J-ring if it satisfies (Ja), (J,) or both of them. A module A is called an essential extension of a submodule B if BnCFCO for every nonzero submodule C of A. A module is said to be injective if it is a direct summand of every extension module. It is well known that every module M has a maximal essential extension ff. A is injective, and is unique to within an isomorphism over M. Let S be a Ji-ring. Then we can define the multiplication in the maximal essential extension 3 of the left S-module S such a way that (i) 3 forms a ring and (ii) the multiplication coincides, on SX3, with the scalar multiplication. This ring is unique up to an isomorphism over S, and is denoted by Sz. As is known, SI is regular (in the sense of von Neumann); and is left self injective, that is, injective as a left module over itself. An extension ring T of a Ji-ring S is called a left quotient ring of S if the left S-module T is an essential extension of the left S-module S. It is also known that every left quotient ring of S is isomorphic, over S, to a subring of 3;. Thus, Sl is the maximal left quotient ring of S. We define similarly a right quotient ring and the maximal right quotient ring ST of a Jr-ring S. For any J-ring S it is easily seen that the following conditions are equivalent: (i) There exists an extension ring T of S with the properties that (a) it is regular (both left and right) self injective, and (b) every nonzero one-sided S-submodule of T has a nonzero intersection with S. (ii) Every left quotient ring of S is a right quotient ring of S, and every right quotient ring of S is a left quotient ring of S. In this case any maximal left quotient ring of S and any maximal

Abstract: Let V be a full Hilbert C*-module over a non unital C*-algebra. Denote by Vd the Hilbert C*-module over the multiplier algebra M(A) consisting of all adjointable maps from A to V. Then V can be naturally embedded in Vd as an ideal submodule and restriction to V gives an isomorphism of C*-algebras of adjointable operators on Vd and on V. The extended module Vd is the completion of V with respect to a variant of strict topology and serves as the largest essential extension of V, thus can be regarded as the Hilbert C*-module version of the multiplier algebra. The extended module Vd of the generalized Hilbert space over A is explicitly determined as a Hilbert C*-module of sequences in M(A) containing the generalized Hilbert space over M(A).