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[2] [a] Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines the use of variables in equations and how to manipulate these equations. [4] [b] Algebra is often understood as a generalization of arithmetic. [8]
Elementary algebra, also known as high school algebra or college algebra, [1] encompasses the basic concepts of algebra. It is often contrasted with arithmetic : arithmetic deals with specified numbers , [ 2 ] whilst algebra introduces variables (quantities without fixed values).
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
Given K-algebras A and B, a homomorphism of K-algebras or K-algebra homomorphism is a K-linear map f: A → B such that f(xy) = f(x) f(y) for all x, y in A. If A and B are unital, then a homomorphism satisfying f(1 A) = 1 B is said to be a unital homomorphism. The space of all K-algebra homomorphisms between A and B is frequently written as
These statements are meaningful once we explain the natural structures of algebra and coalgebra in all the vector spaces involved besides B: (K, ∇ 0, η 0) is a unital associative algebra in an obvious way and (B ⊗ B, ∇ 2, η 2) is a unital associative algebra with unit and multiplication
In universal algebra, an algebraic structure is called an algebra; [2] this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure that is a vector space over a field or a module over a commutative ring.