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Algebra  2013 

On the Jacobson Radical of an -Semiring

DOI: 10.1155/2013/272104

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Abstract:

The notion of -ary semimodules is introduced so that the Jacobson radical of an -semiring is studied and some well-known results concerning the Jacobson radical of a ring (a semiring or a ternary semiring) are generalized to an -semiring. 1. Introduction The concept of semigroups [1] was generalized to that of ternary semigroups [2], that of -ary semigroups [3–6], and even to that of -semigroups [7]. Similarly, it was natural to generalize the notion of rings to that of ternary semirings, that of -ary semirings, and even that of -semirings. Indeed, there were some research articles on semirings, (see, for example, [8–14]), specially on the radical of a semiring; see [15–18]. Semigroups over semirings were studied in [19] and semimodules over semirings were studied in [14]. The notion of semirings can be generalized to ternary semirings [20] and -semirings [21], even to -semirings [22–24]. The radicals of ternary semirings and of -semirings were studied in [20, 21], respectively. The concept of -semirings was introduced and accordingly some simple properties were discussed in [22–24], where the concept of radicals was not mentioned. The notion of the Jacobson radicals was first introduced by Jacobson in the ring theory in 1945. Jacobson [25] defined the radical of , which we call the Jacobson radical, to be the join of all quasi-regular right ideals and verified that the radical is a two-sided ideal and can also be defined to be the join of the left quasi-regular ideals. The concept of the Jacobson radical of a semiring has been introduced internally by Bourne [15], where it was proved that the right and left Jacobson radicals coincide; thus one could say the Jacobson radical briefly. These and some other results were generalizations of well-known results of Jacobson [25]. In 1958, by associating a suitable ring with the semiring, Bourne and Zassenhaus defined the semiradical of the semiring [16]. In [18] it was proved that the concepts of the Jacobson radical and the semiradical coincide. Iizuka [17] considered the Jacobson radical of a semiring from the point of view of the representation theory [15] without reducing it to the ring theory. The external notion of the radical was proved to be related to internal one; at the same time, it was shown that the radical defined in [17] coincides with the Jacobson radical and with the semiradical of the semiring. In the present paper, we investigate -semirings by means of -ary semimodules so that we can define externally the Jacobson radical of an -semiring, and then we establish the radical properties of the

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