In mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative identity element (or unit), i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that e. an element 1 with the property 1x = x1 = x for all elements x of the algebra.

This is equivalent to saying that the algebra is a monoid for multiplication. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation As in any monoid, such a multiplicative identity element is then unique.

Most associative algebras considered in abstract algebra, for instance group algebras, polynomial algebras and matrix algebras, are unital, if rings are assumed to be so. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Most algebras of functions considered in analysis are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with compact support on some (non-compact) space. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set

Given two unital algebras A and B, an algebra homomorphism

f : A → B

is unital if it maps the identity element of A to the identity element of B. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector

If the associative algebra A over the field K is not unital, one can adjoin an identity element as follows: take A×K as underlying K-vector space and define multiplication * by

(x,r) * (y,s) = (xy + sx + ry, rs)

for x,y in A and r,s in K. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Then * is an associative operation with identity element (0,1). The old algebra A is contained in the new one, and in fact A×K is the "most general" unital algebra containing A, in the sense of universal constructions. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

According to the glossary of ring theory, convention assumes the existence of a multiplicative identity for any ring. Ring theory is the branch of Mathematics in which rings are studied that is structures supporting both an Addition and a Multiplication operation In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real With this assumption, all rings are unital, and all ring homomorphisms are unital, and (associative) algebras are unital iff they are rings. ↔ Authors who do not require rings to have identity will refer to rings which do have identity as unital rings, and modules over these rings for which the ring identity acts as an identity on the module as unital modules or unitary modules. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

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