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The number of elements in a minimal system of generators of the homogeneous ideal in the idealization ring

Let (R, m) be a noetherian local ring and M a finitely generated Rmodule. In this paper, we study the number of elements in a minimal system of generators of the homogeneous ideal in the idealization ring RnM. As immediate consequences, we presented a characterization for RnM being a regular local ring and homogeneous ideals of RnM being parameters ideals. | Nguyen Thi Hong Loan, Dao Thi Thanh Ha/ The number of elements in a minimal system. THE NUMBER OF ELEMENTS IN A MINIMAL SYSTEM OF GENERATORS OF THE HOMOGENEOUS IDEAL IN THE IDEALIZATION RING Nguyen Thi Hong Loan, Dao Thi Thanh Ha Natural Education Institute, Vinh University Received on 20/3/2017, accepted for publication on 15/8/2017 Abstract: Let (R, m) be a noetherian local ring and M a finitely generated Rmodule. In this paper, we study the number of elements in a minimal system of generators of the homogeneous ideal in the idealization ring RnM . As immediate consequences, we presented a characterization for RnM being a regular local ring and homogeneous ideals of RnM being parameters ideals. 1 Introduction Throughout this paper R denotes a noetherian local ring with maximal ideal m and M denotes a finitely generated R-module. The concept of principle of idealization was introduced by M. Nagata [9]. We make the Cartesian product R×M become a commutative ring under the componentwise addition and the multiplication defined by (a, x)(b, y) = (ab, ay + bx). This ring is called the idealization of M over R and denoted by RnM. Note that the idealization RnM is again a Noetherian local ring with the unique maximal ideal m × M and dim RnM = dim R. The notion of principle of idealization plays an important role in the study of noetherian rings and modules. Idealization is useful for reducing results concerning submodules to ideals, generalizing results from rings to modules, and constructing examples of commutative rings with zero divisors. For example, Reiten [10] used the principle of idealization to show that any noetherian local ring possessing a Gorenstein module of rank 1 is a homomorphic image of a Gorenstein local ring. Then, Aoyama [2] studied the condition for the idealization to be quasi - Gorenstein and used this for the first step of the proof that any localization of the canonical module K(R) of R at p ∈ SuppR K(R) is the canonical module of .