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 denote...

Abstruct-We consider the class of bilinear time-frequency representations (BTFR's) that are invariant (or covariant) to time shifts, frequency shifts, and time-frequency scalings. This " shift-scale invariant " class is the intersection of the classical shift-invariant (Cohen) class and the recently defined affine class. The mathematical description of shift-scale invariant BTFR's is in terms o...