Narrow Results

Let D be an integral domain, X, Y, U, V be indeterminates over D, D[X, Y, U, V] be the polynomial ring over D, (XV – YU) be the ideal of D[X, Y, U, V] generated by XV – YU, and R = D[X, Y, U, V]/(XV – YU). In this paper, we study some properties of the t-operation on R which turn out to be similar to those of the polynomial ring over D. Among other things, we show that (i) R is a PvMD (resp., ring of Krull type, t-SFT PvMD) if and only if D is, and (ii) if D is a PvMD, then Cl(R) = Cl(D) ⨁ ℤ, where ℤ is the additive group of integers.

Marks called a ring NI if the upper nilradical contains all nilpotent elements. Smoktunowicz constructed an NI ring over which the polynomial ring is not NI. We study some conditions under which polynomial rings are NI. In the process we call a ring Rpolynomial-NI if R[Y] is NI for any finite set Y of commuting indeterminates over R. We show that R is polynomial-NI if and only if R[X] is polynomial-NI for any (possibly infinite) set X of commuting indeterminates over R. It is also shown that R is polynomial-NI if and only if R[x] is NI when R is a ring such that every finite subset of R generates a subring of bounded index of nilpotency, where x is an indeterminate over R.