Commun. Korean Math. Soc. 35 (2020), No. 1, pp. 13–20 https://doi.org/10.4134/CKMS.c180477 pISSN: 1225-1763 / eISSN: 2234-3024 IDEALIZATION OF EM-HERMITE RINGS Hiba Abdelkarim, Emad Abuosba, and Manal Ghanem Abstract. A commutative ring R with unityis called EM-Hermite if for each a, b ∈ R there exist c, d, f ∈ R such that a = cd, b = cf and the ideal (d, f ) is regular in R. We showed in this article that R is a PP-ring if and only if the idealization R(+)R is an EM-Hermite ring if and only if R[x]/(x n+1 ) is an EM-Hermite ring for each n ∈ N. We generalize some results, and answer some questions in the literature. 1. Introduction Let R be a commutative ring with unity. Let Z (R) be the set of zero-divisors in R, and Reg(R)= R\Z (R) be the set of regular elements. An ideal I of R is called a regular ideal if I contains a regular element. A ring R is called EM-Hermite if for each a, b ∈ R, there exist a 1 ,b 1 ,d ∈ R such that a = a 1 d, b = b 1 d and the ideal (a 1 ,b 1 ) is regular. If for each f (x) ∈ Z (R[x]) we can write f (x)= c f f 1 (x), where c f ∈ R and f 1 (x) ∈ reg(R[x]), then R is called an EM-ring, see [1]. It is clear that any EM-Hermite is EM-ring, but the converse is not in general true, see [4]. A ring R is called a morphic ring if for each a ∈ R there exists b ∈ R such that Ann(a)= bR and Ann(b)= aR. It is called generalized morphic if for each a ∈ R there exists b ∈ R such that Ann(a)= bR, see [7]. A ring R is called a PP-ring if every principal ideal of R is a projective R-module. It is well known that R is a PP-ring if and only if for each a ∈ R, Ann(a) is generated by an idempotent if and only if for each a ∈ R there exist an idempotent e and a regular element r such that a = er, see [2]. A ring R is called von Neumann regular if for each a ∈ R there exists b ∈ R such that a = a 2 b. It is well known that R is von Neumann regular if and only if for each a ∈ R there exist an idempotent e and a unit u such that a = eu. A ring R is said to have property A, if a finitely generated ideal I is contained in Z (R) if and only if it has a non-zero annihilator. Received November 17, 2018; Accepted March 13, 2019. 2010 Mathematics Subject Classification. 13A, 13B25, 13B30, 13C10. Key words and phrases. EM-Hermite ring, PP-ring, idealization. This paper is a part of the Ph.D. thesis of the first author under supervision of the other authors in The University of Jordan. c 2020 Korean Mathematical Society 13