Extensions and topological conditions of NJ rings

Abstract

A ring R is said to be NJ if J(R) = N (R) . This paper mainly studies the relationship between NJ rings and related rings, and investigates the Dorroh extension, the Nagata extension, the Jordan extension, and some other extensions of NJ rings. At the same time, we also prove that if R is a weakly 2-primal α -compatible ring with an isomorphism α of R , then R[x; α] is NJ; if R is a weakly 2-primal δ -compatible ring with a derivation δ of R , then R[x; δ] is NJ. Moreover, we consider some topological conditions for NJ rings and show for a NJ ring R that R is J-pm if and only if J -Spec(R) is a normal space if and only if M ax(R) is a retract of J -Spec(R) .

$Σ$-semicommutative rings and their skew PBW extensions

“…According to Proposition 5.3, if R satisfies any of the give conditions, then A is an NI ring. Thus, [55,Theorem 4.10] implies that A is NJ, whence the statements are equivalent by [27,Theorem 4.10].…”

“…It is clear that NJ and 2-primal rings are NI. However, in [27,Example 2.2] an example of a ring that is 2-primal but not NJ is presented, and we can also find an NJ ring that it is not 2-primal [26,Example 1.2]. Shin [53] and Sun [58] studied some topological conditions for 2-primal rings.…”

“…Shin [53] and Sun [58] studied some topological conditions for 2-primal rings. Hwang et al [26] and Jiang et al [27] studied topological conditions for NI and NJ rings, respectively.…”

On NI and NJ skew PBW extensions

“…Jiang et al, [17] studied the relationship between NJ rings and some families of rings. They investigated extensions as Dorroh, Nagata, and Jordan.…”

“…Theorem 3.12 (1)). Moreover, they considered some topological conditions for NJ rings and showed relations between algebraic and topological notions ( [17], Section 4).…”