Abstract
Rings considered in this article are commutative with identity. A subring of a ring is assumed to contain the identity element of the ring. Let S be a multiplicatively closed subset of a ring R satisfying the following property (P): whenever \({ab \in S}\) with at least one of a, b is in S, then both of them are in S. The (P)-closure of any multiplicatively closed subset of R is discussed; and saturated multiplicatively closed subsets of the form 1 + I, where I is an ideal of R are also considered. We investigate the question of determining rings admitting only a finite number of such multiplicatively closed sets. In addition, we discuss the problem of whether a ring R admitting only a finite number of saturated multiplicatively closed subsets of the form 1 + I, where I is an ideal of R, extends to the polynomial ring R[x] in one variable x over R.
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Visweswaran, S. Some remarks on multiplicatively closed sets. Arab. J. Math. 2, 409–425 (2013). https://doi.org/10.1007/s40065-013-0078-x
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DOI: https://doi.org/10.1007/s40065-013-0078-x