By A.I. Kostrikin, I.R. Shafarevich, E. Behr, Yu.A. Bakhturin, L.A. Bokhut, V.K. Kharchenko, I.V. L'vov, A.Yu. Ol'shanskij

The algebra of sq. matrices of dimension n ~ 2 over the sphere of complicated numbers is, obviously, the best-known instance of a non-commutative alge 1 bra • Subalgebras and subrings of this algebra (for instance, the hoop of n x n matrices with necessary entries) come up certainly in lots of parts of mathemat ics. traditionally despite the fact that, the learn of matrix algebras was once preceded through the invention of quatemions which, brought in 1843 via Hamilton, stumbled on ap plications within the classical mechanics of the earlier century. Later it became out that quaternion research had vital purposes in box conception. The al gebra of quaternions has develop into one of many classical mathematical items; it truly is used, for example, in algebra, geometry and topology. we are going to in short specialise in different examples of non-commutative earrings and algebras which come up evidently in arithmetic and in mathematical physics. the outside algebra (or Grassmann algebra) is usual in differential geometry - for instance, in geometric thought of integration. Clifford algebras, which come with external algebras as a unique case, have purposes in rep resentation conception and in algebraic topology. The Weyl algebra (Le. algebra of differential operators with· polynomial coefficients) usually looks within the illustration idea of Lie algebras. in recent times modules over the Weyl algebra and sheaves of such modules turned the root of the so-called microlocal research. the idea of operator algebras (Le.

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**Example text**

Cn[x] is a semivector space over Cn. 9: Let V = Cn × Cn × Cn, V is a semivector space over Cn. It is left for the reader to verify. Let S be a semifield and V be a semivector space over S. If β = ∑ α i v i (vi ∈ V, αi ∈ S) which is in V, we use the terminology β is a linear combination of vi's. We also say β is linearly dependent on vi's if β can be expressed as a linear combination of vi's. We see the relation is a non-trivial relation if at least one of the coefficients αi's is non zero. This set {v1, v2, … , vk} satisfies a non trivial relation if vj is a linear combination of {v1, v2, … , vj-i, vj+i, … , vk}.

Thus we can easily verify all lattices L whether L is a distributive lattice or otherwise L is a semivector space over the semifield C2. 10: A subsemivector space W of a semivector space V over a semifield S is a non-empty subset of V, which is itself, a semivector space with respect to the operations of addition and scalar multiplication. Remark: The whole space V and the {0} element are trivially the subsemivector spaces of every semivector space V. 15: Ro is a semivector space over Zo. Qo the subset of Ro is a non-trivial subsemivector space of Ro.

A set of vectors, which is not linearly dependent, is called linearly independent. 17: Let V = V1 ∪ V2 be a bisemivector space over the semifield F. A linearly independent set P = A ∪ B spanning the bisemivector space V is called the basis of V. 24: Let V = V1 ∪ V2, where V1 = Zo × Zo and V2 = Z 8o [ x ] . Clearly V is a bisemivector space over Zo and the unique basis for this bisemivector space is P = {(1, 0), (0, 1)} ∪ {1, x, x2, … , x8}. Z 8o [ x ] denotes the collection of all polynomials of degree less than or equal to 8 with coefficient from Zo.