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Thus y = x + w ∈ x + H , and y − x ∈ H ⇐⇒ −(y − x) = x − y ∈ H ⇐⇒ x ∈ y + H . In synthesis, y ∈ x + H ⇐⇒ y − x ∈ H ⇐⇒ x ∈ y + H. Finally, we saw p : G → G/H given by x −→ x + H. If x, w ∈ G , then p(x + w) = (x + w) + H = (x + H) + (w + H) = p(x) + p(w). Hence, p is a homomorphism called the canonical projection. All of this was done for vector spaces over a field K . However, recall that the additive part of a vector space is an abelian group. What about the non-commutative case? What happens?

9 Example. (Mn K, +, ·, µ), where Mn K denotes the n × n square matrices with coefficients in a field K ( µ denotes the scalar multiplication) is an algebra, the same as (K, +, ·, µ) and (K[x], +, ·, µ) . ) of algebras Ai , one for each index i ∈ N. For those who have studied, in an elementary Linear Algebra, or Multilinear Algebra course, recall the following concepts from Multilinear Algebra (as in [Ll2]), that are not requisites for this text. 10 Example. Let T k (V ) = ⊗k V = V ⊗K · · · ⊗K V be the tensorial product of a vector space V on the field K, k times.

If G is abelian, we have n(xy) = (xy)n = xn y n = (nx)(ny). Hence, every abelian group G can be seen as a group with operators in Z . 8 Definition. A ring is a triple (Λ, +, ·) where Λ is a set, + and · are binary operations such that 1. (Λ, +) is a commutative group, 2. (Λ, ·) is a semigroup, 3. u(v + w) = uv + uw and (u + v)w = uw + vw . The reader can show that (Z, +, ·) , (Zn , +, ·) , (Q, +, ·) , (R, +, ·) , (Mn K, +, ·) , (K, +, ·) , (K[x], +, ·) , (C, +, ·) are rings. If a ring (Λ, +, ·) satisfies: 4.

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