Right -Weakly Regular -Semirings

The concepts of a -idempotent -semiring, a right -weakly regular -semiring, and a right pure -ideal of a -semiring are introduced. Several characterizations of them are furnished. 1. Introduction -semiring was introduced by Rao in [1] as a generalization of a ring, a -ring, and a semiring. Ideals in semirings were characterized by Ahsan in [2], Iséki in [3, 4], and Shabir and Iqbal in [5]. Properties of prime and semiprime ideals in -semirings were discussed in detail by Dutta and Sardar [6]. Henriksen in [7] defined more restricted class of ideals in semirings known as -ideals. Some more characterizations of -ideals of semirings were studied by Sen and Adhikari in [8, 9]. -ideal in a -semiring was defined by Rao in [1] and in [6] Dutta and Sardar gave some of its properties. Author studied -ideals and full -ideals of -semirings in [10]. The concept of a bi-ideal of a -semiring was given by author in [11]. In this paper efforts are made to introduce the concepts of a -idempotent -semiring, a right -weakly regular -semiring, and a right pure -ideal of a -semiring. Discuss some characterizations of a -idempotent -semiring, a right -weakly regular -semiring, and a right pure -ideal of a -semiring. 2. Preliminaries First we recall some definitions of the basic concepts of -semirings that we need in sequel. For this we follow Dutta and Sardar [6]. Definition 1. Let and be two additive commutative semigroups. is called a -semiring if there exists a mapping denoted by , for all and satisfying the following conditions: (i); (ii); (iii); (iv), for？？all and for all . Definition 2. An element 0 in a -semiring is said to be an absorbing zero if , , for all and . Definition 3. A nonempty subset of a -semiring is said to be a sub--semiring of if is a subsemigroup of and , for all and . Definition 4. A nonempty subset of a -semiring is called a left (resp., right) ideal of if is a subsemigroup of and ) for all , and . Definition 5. If is both left and right ideal of a -semiring , then is known as an ideal of . Definition 6. A right ideal of a -semiring is said to be a right -ideal if and such that ; then . Similarly we define a left -ideal of a -semiring . If an ideal is both right and left -ideal, then is known as a -ideal of . Example 7. Let denote the set of all positive integers with zero. is a semiring and with , forms a -semiring. A subset of is an ideal of but not a -ideal. Since and but . Example 8. If is the set of all positive integers, then (, max., min.) is a semiring and with , forms a -semiring. is a -ideal for any . Definition 9. For a nonempty of a