
Algebra 2014
Right Weakly Regular SemiringsDOI: 10.1155/2014/948032 Abstract: 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 biideal 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 subsemiring 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
