It is well known that if delta : S -> S is a derivation in semiring S, then in the semiring Mn(S) of n x n matrices over S, the map delta(her) such that delta(her)(A) = (delta(a(ij))) for any matrix A = (a(ij)) is an element of M-n(S) is a derivation. These derivations are used in matrix calculus, differential equations, statistics, physics and engineering and are called hereditary derivations. On the other hand (in sense of [Basic Algebra II (W. H. Freeman & Company, 1989)]) S-derivation in matrix semiring Mn(S) is a S-linear map D : M-n(S) -> M-n(S) such that D(AB) = AD(B) + D(A)B, where A,B is an element of M-n(S). We prove that if S is a commutative additively idempotent semiring any S-derivation is a hereditary derivation. Moreover, for an arbitrary derivation delta(her) the derivation delta in S is of a special type, called inner derivation (in additively, idempotent semiring). In the last section of the paper for a noncommutative semiring S a concept of left (right) Ore elements in S is introduced. Then we extend the center C(S) to the semiring LO(S) of left Ore elements or to the semiring RO(S) of right Ore elements in S. We construct left (right) derivations in these semirings and generalize the result from the commutative case.
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