Let W b e a class of not necessarily associative rings which is universal in the sense that it is closed under homomorphic images and is hereditary to subrings. All rings considered will be assumed to belong to W. The notation I < R will mean / is an ideal of R. A relation a on W will be called an H-relation if a satisfies the properties:(1) IcrR implies / is a subring of R. Examples of //-relations are "subring of", "left ideal of" and "ideal of". A large class of examples is provided by the following propositions, the proofs of which are elementary.
Proposition 1. Let M C W be closed under homotnorphisms and hereditary to ideals. Then the following are H-relations: (i) {(/, R)\I is an ideal of R and R/I G M}. (ii) {(/, R)\I is a subring of R and I G M}. (iii) {(/, R)\I is a subring of R and the ideal G(I, R) of R generated by I is in M}.
Proposition 2. Any union or intersection of H-relations is again an H-relation.
Thus for example {(/, R)\I is a commutative left ideal of R} and {(/, R)\ I is an ideal of finite index in
Proposition 3. If a and T are H-relations, then cr ° T and a A T are also H-relations.For example, let a = "is an accessible subring of"; then a = U "=i<3 ". If M is any subclass of W we denote by LM the lower radical class determined in W by M. It is proved in (3) that if M is hereditary to ideals, so is LM. This result is reproved in (4), and it is observed in (7) that with
Various properties possessed by a class M of not necessarily associative rings are inherited by the lower radical LM, including heredity to one-sided ideals or subrings, failure to contain nonzero noetherian rings and, in the associative case, failure to contain nonzero artinian rings.
8This paper considers how the use o f AWK as a programming vehicle may clarif y certain aspects of the traditional maste r file update problem . We view the updat e problem as having one input stream, rathe r than two separate streams, of old maste r records and transaction records . Thi s permits us to see the programming proble m as the implementation of decision-tabl e logic .For definiteness, let us pose a sampl e master file update problem that has a small scale . In the following, we separat e fields of the records by using a colon , as a delimiter . We have a file o f master records . Suppose that thes e represent inventory records, and consis t of a five-digit item number, an 'M' fo r master record (we shall see T
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