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A generalization of prime ideals in rings

Kentaro Murata, Yoshiki B. Kurata, Hidetoshi Marubayashi

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Abstract

In [2], van der Walt has defined s-prime ideals in noncommutative rings and obtained analogous results of McCoy [1] for s-prime ideals. In the present paper, we shall give a generalized concept of prime ideals, called /-prime ideals, by using some family of ideals, and obtain analogous results in [2], If our family of ideals is, in particular, the set of principal ideals of the ring, the /-prime ideals coincide with the prime ideals and conversely. In addition, if we take multiplicatively closed systems as kernels, the /-prime ideals coincide with the s-prime ideals.

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In [2], van der Walt has defined s-prime ideals in noncommutative rings and obtained analogous results of McCoy [1] for s-prime ideals. In the present paper, we shall give a generalized concept of prime ideals, called /-prime ideals, by using some family of ideals, and obtain analogous results in [2], If our family of ideals is, in particular, the set of principal ideals of the ring, the /-prime ideals coincide with the prime ideals and conversely. In addition, if we take multiplicatively closed systems as kernels, the /-prime ideals coincide with the s-prime ideals.

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Available abstract

In [2], van der Walt has defined s-prime ideals in noncommutative rings and obtained analogous results of McCoy [1] for s-prime ideals. In the present paper, we shall give a generalized concept of prime ideals, called /-prime ideals, by using some family of ideals, and obtain analogous results in [2], If our family of ideals is, in particular, the set of principal ideals of the ring, the /-prime ideals coincide with the prime ideals and conversely. In addition, if we take multiplicatively closed systems as kernels, the /-prime ideals coincide with the s-prime ideals.

Key concepts: Mathematics, Boolean prime ideal theorem, Prime (order theory), Associated prime, Semiprime ring, Prime element, Fractional ideal, Generalization

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