Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$. In the case that $\mathrm{k}$ possesses $3$-class $C_{\mathrm{k},3}$ type $(9,3)$, capitulation $3$-ideal classes in its unramified cyclic cubic extensions determine...

We recall that a ring R is called near pseudo-valuation ring if every minimal prime ideal is a strongly prime ideal. Let R be a commutative ring, σ an automorphism of R. Recall that a prime ideal P of R is σ-divided if it is comparable (under inclusion) to every σ-stable ideal I of R. A ring R is called a σ-divided ring if every prime ideal of R is σ-divided. Also a ring R is almost σ-divided r...

Throughout, all rings are associative with an identity element but not necessarily commutative. During the decade of 1980s, a series of papers appeared in the Canadian Journal of Mathematics [1, 2] that had investigated pairs of commutative rings with the same set of prime ideals. In this paper, we consider some generalizations of that study in the noncommutative setting. Consider the ring R= H...