Unitary strongly prime rings and related radicals

Unitary Strongly Prime Rings and Related Radicals

A unitary strongly prime ring is defined as a prime ring whose central closure is simple with identity element. The class of unitary strongly prime rings is a special class of rings and the corresponding radical is called the unitary strongly prime radical. In this paper we prove some results on unitary strongly prime rings. The results are applied to study the unitary strongly prime radical of...

Prime Decompositions of Radicals in Polynomial Rings

In the last twenty years several methods for computing primary decompositions of ideals in multivariate polynomial rings over fields (Seidenberg (1974), Lazard (1985), Kredel (1987), Eisenbud et al. (1992)), the integers (Seidenberg, 1978), factorially closed principal ideal domains (Ayoub (1982), Gianni et al. (1988)) and more general rings (Seidenberg, 1984) have been proposed. A related prob...

Prime Radicals of Skew Laurent Polynomial Rings

Let R be a ring with an automorphism σ. An ideal I of R is σ-ideal of R if σ(I) = I. A proper ideal P of R is σ-prime ideal of R if P is a σ-ideal of R and for σ-ideals I and J of R, IJ ⊆ P implies that I ⊆ P or J ⊆ P . A proper ideal Q of R is σ-semiprime ideal of Q if Q is a σ-ideal and for a σ-ideal I of R, I2 ⊆ Q implies that I ⊆ Q. The σ-prime radical is defined by the intersection of all ...

Prime Ideals and Strongly Prime Ideals of Skew Laurent Polynomial Rings

Strongly nil-clean corner rings

We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings‎, ‎then $R/J(R)$ is nil-clean‎. ‎In particular‎, ‎under certain additional circumstances‎, ‎$R$ is also nil-clean‎. ‎These results somewhat improves on achievements due to Diesl in J‎. ‎Algebra (2013) and to Koc{s}an-Wang-Zhou in J‎. ‎Pure Appl‎. ‎Algebra (2016)‎. ‎...