Locally Compact Baer Rings

Locally direct sums [W, Definition 3.15] appeared naturally in classification results for topological rings (see, e.g.,[K2], [S1], [S2], [S3], [U1]). We give here a result (Theorem 3) for locally compact Baer rings by using of locally direct sums. 1. Conventions and definitions All topological rings are assumed associative and Hausdorff. The subring generated by a subset A of a ring R is denoted by A . A semisimple ring means a ring semisimple in the sense of the Jacobson radical. A non-zero idempotent of a ring R is called local provided the subring eRe is local. The closure of a subset of a topological space X is denoted by A . The Jacobson radical of a ring R is denoted by J(R). A compact element of a topological group [HR, Definition (9.9)] is an element which is contained in a compact subgroup. The symbol ω stands for the set of all natural numbers. All necessary facts concerning summable families of elements of topological Abelian groups can be found in [W, Chapter II,10, pp.71-80]. If R is a ring, a∈R, then a={x∈R: ax=0}. Recall that a ring R with identity is called a Baer ring if for each a∈R, there exists a central idempotent ε such that a=Rε. The following properties of a Baer ring are known: i) Any Baer ring does not contain non-zero nilpotent elements. Indeed, let a∈R, a2=0. Let a=Rε, where ε is a central idempotent of R. Then a=aε=0. ii) If R is a Baer ring, a,b∈R, n a positive natural number and ba=0, then ba=0. Indeed, b1ab=0, hence b1aba=0. Continuing, we obtain that (ba)=0, hence ba=0. Recall [K1,p.155] that a topological ring R is called a Q-ring provided the set of all quasiregular elements of R is open (equivalently, R has a neighbourhood of zero consisting of quasiregular elements). Definition 1. A topological ring R is called topologically strongly regular if for each x∈R there exists a central idempotent e such that Re Rx = . Mihail Ursul-Locally compact Baer rings 220 We note that a topologically strongly regular ring has no non-zero nilpotent elements. Let { α R }α∈Ω be a family of topological rings, for each α∈Ω let Sα be an open subring of Rα. Consider the Cartesian product ∏ Ω ∈ α α R and let A={{ }∈ α x ∏ Ω ∈ α α R : xα∈Sα for all but finitely many α∈Ω}. The neighborhoods of zero of ∏ Ω ∈ α α S endowed with its product topology form a fundamental system of neighborhoods of zero for a ring topology on A. The ring A with this topology is called the local direct sum [W,Definition 31.5] of { α R }α∈Ω relative to { α S }α∈Ω and is denoted by ( ) ∏ Ω ∈ α α α S R : . Definition 2. A topological ring R is called a S-ring if there exists a family { α R }α∈Ω of locally compact division rings with compact open subrings α S with identity such that R is topologically isomorphic to the locally direct product ( ) ∏ Ω ∈ α α α S R : . We will say that an element x of a topological ring R is discrete provided the subring Rx is discrete.