It is well known that if δ : S → S is a derivation in semiring S, then in the semiring Mn(S) of n × n matrices over S, the map δher such that δher(A) = (δ(aij)) for any matrix A = (aij) ∈ Mn(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 : Mn(S) → Mn(S) such that D(AB) = AD(B) + D(A)B, where A,B ∈ Mn(S). We prove that if S is a commutative additively idempotent semiring any S-derivation is a hereditary derivation. Moreover, for an arbitrary derivation δher the derivation δ 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.

中文翻译：

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矩阵半环的推导

众所周知，如果δ ： 小号 → 小号是半环的导数小号，然后在半环米n(小号)的n × n矩阵小号， 地图δ她这样δ她(一种) = (δ(一种一世j))对于任何矩阵一种 = (一种一世j) ∈ 米n(小号)是一个推导。这些推导用于矩阵微积分、微分方程、统计学、物理学和工程学，被称为遗传推导。另一方面（在 [基础代数二（WH 弗里曼公司，1989 年）]）小号-矩阵半环中的推导米n(小号)是一个小号- 线性地图D ： 米n(小号) → 米n(小号)这样D(一种乙) = 一种D(乙) + D(一种)乙， 在哪里一种,乙 ∈ 米n(小号). 我们证明如果小号是可交换的加性幂等半环小号-派生是遗传派生。此外，对于任意推导δ她推导δ在小号是一种特殊类型，称为内推导（在加法中，幂等半环）。在论文的最后一节中，关于非交换半环小号左（右）矿石元素的概念小号介绍。然后我们扩展中心C(小号)到半环 LO(小号)左矿石元素或半环 RO(小号)中的正确矿石元素小号. 我们在这些半环中构造左（右）导数，并推广交换情况的结果。