^{1} and here we extend this concept to theory of semirings. We attempt to induce commutativity in weakly cancellative semirings ^{2} whose concept is unorthodox in the theory of semirings. This article pave new ways to study derivations and its applications on semirings.

The formal definition of semirings was introduced by H. S. Vandiver in 1934 and has since then been studied by many authors. Basic references for semirings are ^{3, 4}. Nowadays semirings have important applications in the theory of automata, formal languages and in theoretical computer science (cf. ^{4, 5}). In connection with these facts, semirings are studied in various directions. One of the main goals is the study of mappings on semirings and it’s relationships with commutativity of semirings (cf. ^{6, 2, 7, 8}).

An additive mapping d: R → R is called a derivation of ring R if d(xy) = xd(y) + d(x)y holds for all x,y ∈ R. A map d: R → R (may not be additive) is called a multiplicative derivation on R if d(xy) = xd(y) + d(x)y holds for all x, y ∈ R. In the year 1991, Bre˘sar ^{9} introduced the concept of generalized derivation in rings as follows; an additive mapping G: R → R associated with derivation d: R → R such that G(xy) = G(x)y + xd(y) holds for all x, y ∈ R. It is obvious that any derivation is a generalized derivation, but the converse is not true in general. The notion of derivation has been generalized in several ways by various authors in rings ^{10, 11, 12, 13}. We introduce the concept of generalized m-derivations in^{1} as an additive map G : R → R on ring R and there exists a multiplicative derivation d of R such that G(xy) = G(x)y + xd(y) for all x, y ∈ R. Here we extend the concept of generalized m-derivations in semirings. To elaborate completely, we have provided some examples (see Examples). During the last few years, there has been ongoing interest concerning the relationship between the commutativity of a ring R and the existence of derivations of R. In the sequel, Martindale ^{14} used derivations to enforce commutativity in rings, therefore, it remains notable for many researchers and they attempt to extend it with generalized derivations satisfying certain polynomial constraints., we refer the reader to see ^{10, 11, 12, 13, 1} for further references. This type of researches provides us motivation to extend some remarkable theorems of commutativity on generalized m-derivations in semirings. Moreover, we study the characteristics of composition of generalized m-derivations in semirings.

An element a ∈ S is said to be additively left (resp. right) cancellable if a + b = a + c (resp. b + a = c + a) yields b = c. In this paper, semiring means additively cancellative semiring. A semiring S is said to be weakly left cancellative (WLC) if axb = axc for all x ∈ S, implies either a = 0 or b = c. This notion is recently introduced by V. De. Fillips ^{2}. In the same pattern, we extend this concept to weakly right cancellative (WRC) semirings if bxa = cxa for all x ∈ S, implies either a = 0 or b = c. A semiring which is weakly left as well as right cancellation is termed as weakly cancellative (WC) semiring. We will denote Z(R) = {z ∈ S : zx = xz, ∀x ∈ S} the center of S, and x ◦ y = xy + yx the Jordan product of x, y ∈ R. Throughout this paper, R will represent a weakly cancellative semiring (WC) with center Z(R). A semiring R is said to be 2-torsion free if 2x = 0, for x ∈ R, implies that x = 0.

The following examples confirm the existence of such maps in semirings.

where D is distributibe lattice, then R is additively inverse semirings. The additive inverse element a’=a, for all a ∈ R. Now define G, g: R→R as follows:

This can be verified that g is multiplicative derivation and G is generalized m-derivations.

where S is additively inverse semirings, then R is additively inverse semirings. Define g, G: R→R as follows:

Here g is multiplicative derivation which is non-additive and G is generalized m-derivations.

In this section, we extend the idea of H.E. Bell and Martindale ^{14} to enforce commutativity in semirings by using generalized m-derivations.

Proof: We have

Replacing y by yx in (1), we have

Using (1), we have

Substituting

Multiplying r from right in (3) gives

From (4) and (5), we have

This implies that either

Let

By using (6) twice, we get

On replacing

By the assumption, we have

This implies that

If

Proof. We have

Replacing

By adding

Replacing

Further, (4) also gives that

The last two expressions yields that

Hence, we get that either

From above two theorems, we get the following corollary as their immediate consequence:

In this section, we find the result that if composition of two generalized m-derivations are also generalized m-derivation then one of their associated derivation must be zero. For this result the following lemma is crucial.

On replacing x with xy, we get

By using (1), we get

or

By using the weakly cancellation property of

Let be a 2-torsion free prime ring and

Moreover,

From (1) and (2) and R is WC semirings therefore we get

On substituting

while (3) implies that

Now replace

By using (4), we get

By using the primeness of R, we obtain

Now replace x by

By taking

or

By using (3), we get

Now adding

The generalized m-derivations, have great potential to play an important role in the theory of semiring as well as other fields of mathematics (like theory of semirings with Involutions, functional analysis, C^{*}− algebras, B^{*} − algebras and linear differential equations etc). This article explores new way to study the commutativity of semirings through generalized m-derivations. The questions arise here is interesting to discuss that how the generalized m-derivations induce commutativity in semirings with involution. This article invites the researcher’s to explore the results, discussed here, in the canvas of ideals in semirings.

The authors are grateful to the anonymous reviewer for his/her critical comments to improve the quality of the manuscript.