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A proper ideal I of a commutative ring R is said to be a strongly quasi-primary ideal if, whenever a , b ∈ R with a b ∈ I , then a 2 ∈ I or b ∈ I (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of R , called the class of quasi- n -ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of R is a maximal or a minimal ideal of R. Many examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper we investigate identities with two generalized derivations in prime rings. Let R be a 2-torsion free prime ring admitting two generalized derivations F and G, not both zero. Among others, we prove that if F(xy) + G(yx) 2 Z(R) for all x; y 2 R, then R is a commutative. Also, if the ring R is equipped with an involution of the second kind and F(xx) + G(x+x) 2 Z(R) for all x 2 R, then R is commutative. The proved theorems give a rise to many corollaries which recover well-known results on (generalized) derivations and left multiplier maps on prime rings (resp. with involution). All along the paper, examples are given to discuss the necessity of our assumptions. [ABSTRACT FROM AUTHOR]

Let A be a commutative ring and I an ideal of A. The amalgamated duplication of A along I, denoted by $${A \bowtie I}$$ , is the special subring of $${A \times A}$$ defined by $${A \bowtie I } := \pi \times_{\frac{A}{I}} \pi = \{(a, a + i) \mid a \in A, i \in I\}$$ . We are interested in some basic and homological properties of a special kind of $${A \bowtie I}$$ -modules, called the duplication of M along I with M is an A-module, and defined by $${M \bowtie I := \{(m, m') \in M \times M \mid m - m^{\prime} \in IM\}}$$ . The new results generalize some results on amalgamated duplication of a ring along an ideal. [ABSTRACT FROM AUTHOR]

In this note, we characterize the (weak) Gorenstein global dimension for arbitrary associative rings. Also, we extend the well-known Hilbert's syzygy Theorem to the weak Gorenstein global dimension, and we study the weak Gorenstein homological dimensions of direct product of rings which gives examples of non-coherent rings with finite Gorenstein dimensions > 0 and infinite classical weak dimension. [ABSTRACT FROM AUTHOR]