One Counterexample for Two Open Questions about the Rings R(X) and R<X> Page 1 Ž .
Journal of Algebra 222, 424427 1999 doi:10.1006rjabr.1999.8009, available online at http:rrwww.idealibrary.com …

We construct two counterexamples to the open questions: is R〈 n〉 strong S (resp. catenary)
when R (n) is? The first example is a ring R such that R (n) is strong S and R〈 n〉 is not. The …

A module M is called a CS-module if every submodule of M is essential in a direct summand
of M. A ring R is called CS-semisimple if every right R-module is CS. For a ring R, we show …

We determine entirely which Artinian rings have maximal subring. In particular, we show that
an Artinian ring without maximal subring is integral over some finite subring and in particular …

A ring R is called a right Harada ring if it is right Artinian and every non-small right R-module
contains a non-zero injective submodule. The first result in our paper is the following: Let R …

Let R be a ring, R is called a right ACS ring if the right annihilator of each element is
essential in a direct summand of R. We prove that right CF ring is right artinian under right …

A result of Nakayama and Skornyakov states that a ring R is an Artinian serial ring if and
only if every R-module is serial. This motivated us to study commutative rings for which every …

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition (r.
RMC r. RMC, for short) if R/A is an Artinian right R-module for any essential right ideal A of …

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if
and only if (i) for any nonzero right module, there is a nonzero linear map from it to a …

Let R and S be commutative rings with identity, f: R→ S a ring homomorphism and J an ideal
of S. Then the subring R⋈ f J:={(r, f (r)+ j)| r∈ R and j∈ J} of R× S is called the amalgamation …