By Fosner A., Fosner M.

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Iv. v. vi. (D, +, 0) is a loop under ‘+’. a + b = 0 implies b + a = 0 for all a, b ∈ D. (D∗, y, 1) is a group where D∗ = D\ {0}. 0 y a = a y 0 = 0 for all a ∈ D. a y (b + c) = a y b + a y c for all a, b, c ∈ D. For every pair a, b ∈ D there exists da, b ∈ D∗ such that for every x ∈ D; a + (b + x) = (a + b) + da, b x. Now [118 and 126] has defined loop near domains analogous to group rings. DEFINITION [118]: Let L be a finite loop under ‘+’ and D be a near domain, the loop near domain DL contains elements generated by di mi where di ∈ D and mi ∈ L where we admit only finite formal sums satisfying the following: i.

For 0 ∈ S and α ∈ V we have 0 yα = 0. To every scalar s ∈ S and every vector v ∈ V there is associated a unique vector called the product s y v, which we denote by sv. Scalar multiplication is associative (ab) α = a(bα) for all a, b ∈ S and α ∈ V. Scalar multiplication distributes, that is a (α + β) = aα + aβ for all a ∈ S and α, β ∈ V. Scalar multiplication is distributive with respect to scalar addition: (a + b)α = aα + bα for all a, b ∈ S and for all α ∈ V. 1yα = α (where 1 ∈ S) and α ∈ V. 6: Zo is a semifield and Zo [x] is a semivector space over Zo.

67: Let R be a ring. An element x ∈ A ⊆ R where A is a S-subring of R is said to be a Smarandache quasi semicommutative (S-quasi semicommutative) if there exists y ∈ A (y ≠ 0) such that xy – yx commutes with every element of A. If x ∈ A ⊂ R is a Smarandache semicommutator (S-semicommutator) of x denoted by SQ (x) = {p ∈ A / xp – px commutes with every element of A}; R is said to be a Smarandache quasi semicommutative ring (S-quasi semicommutative ring) if for every element in A is S-quasi semicommutative.

### 2-local superderivations on a superalgebra Mn(C) by Fosner A., Fosner M.

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