Available online at http://scik.org Algebra Letters, 2014, 2014:1 ISSN: 2051-5502 2-PRIMAL WEAK (σ , δ )-RIGID RINGS M. ABROL, V. K. BHAT * School of Mathematics, SMVD University, P/o SMVD University, Katra, J and K, India Copyright c  2014 Abrol and Bhat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. For a ring R, an endomorphism σ of R and δ a σ -derivation of R, we introduce a weak (σ , δ )-rigid ring, which generalizes the notion of (σ , δ )-rigid rings and investigate its properties. Moreover, we state and prove a necessary and sufﬁcient condition for a weak (σ , δ )-rigid ring to be a (σ , δ )-rigid ring. We prove that a (σ , δ )-ring is a weak (σ , δ )-rigid ring and conversely that the prime radical of a weak(σ , δ )-rigid ring is a (σ , δ )-ring. We also ﬁnd a relation between minimal prime ideals and completely prime ideals of a ring R, where R is a (σ , δ )-ring and R is a 2-primal weak (σ , δ )-rigid ring. Keywords: minimal prime ideals, completely prime ideals, (σ , δ )-rings, weak (σ , δ )-rigid rings, 2-primal rings. 2010 AMS Subject Classiﬁcation: 16S36, 16N40, 16P40. 1. Introduction A ring R always means an associative ring with identity 1 6= 0, unless otherwise stated. The prime radical and the set of nilpotent elements of R are denoted by P(R) and N (R) respectively. The ring of integers is denoted by Z, the ﬁeld of real numbers is denoted by R, the ﬁeld of rational numbers is denoted by Q and the ﬁeld of complex numbers is denoted by C, unless otherwise stated. The set of minimal prime ideals of R is denoted by Min.Spec(R). * Corresponding author Received March 28, 2014 1