
Algebra 2013
On the Jacobson Radical of an SemiringDOI: 10.1155/2013/272104 Abstract: The notion of ary semimodules is introduced so that the Jacobson radical of an semiring is studied and some wellknown 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 quasiregular right ideals and verified that the radical is a twosided ideal and can also be defined to be the join of the left quasiregular 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 wellknown 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
