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– 276 . [7] B akkari , C. , K abbaj , S. and M ahdou , N. , Trivial extension definided by Prûfer conditions , J. Pure Appl. Algebra , 214 ( 2010 ), 53 – 60

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Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IMN. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

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Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(x n) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

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. Math. , 83 ( 1979 ), 375 - 379 . [12] Kabbaj , S. and Mahdou , N. , Trivial extensions defined by coherent-like conditions , Comm. Algebra , 32 ( 10 ) ( 2004

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. Proof Trivial extension of the proof of Proposition 1. Reference classes We now turn to the definition and implementation of reference classes. For this purpose, we first

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