Some properties of associate and presimpliﬁable rings

In this paper we study some properties of associate and presimpliﬁable rings. We give a characterization of the associate (resp., domainlike) pullback P of R1 → R3 ← R2 ,where R1 and R2 are two presimpliﬁable (resp., domainlike) rings. We prove that R is presimpliﬁable ring if and only if the factor ring R/nil(R)is presimpliﬁable and the ideal nil(R) is presimpliﬁable. Then we investigate the associate and presimpliﬁable property of the dual rings $R[x]/langle x^2 rangle$ and its modules through the base ring R and its modules.

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