The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a module or vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over K. A standard first example of a K-algebra is a ring of square matrices over a commutative ring K, with the usual matrix multiplication.
In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras.
Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A.
By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map The associativity then refers to the identity: An associative algebra amounts to a ring homomorphism whose image lies in the center.
If A is an R-algebra, taking x = 1, the same formula in turn defines a ring homomorphism η : R → A whose image lies in the center.
The ring homomorphism η appearing in the above is often called a structure map.
In the commutative case, one can consider the category whose objects are ring homomorphisms R → A for a fixed R, i.e., commutative R-algebras, and whose morphisms are ring homomorphisms A → A′ that are under R; i.e., R → A → A′ is R → A′ (i.e., the coslice category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R. How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry.
For example, take A to be the ring of continuous functions on a compact group G. Then, not only A is an associative algebra, but it also comes with the co-multiplication Δ(f)(g, h) = f(gh) and co-unit ε(f) = f(1).
[1] The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom.
The fact that A is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of A is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)
The Wedderburn principal theorem states:[6] for a finite-dimensional algebra A with a nilpotent ideal I, if the projective dimension of A / I as a module over the enveloping algebra (A / I)e is at most one, then the natural surjection p : A → A / I splits; i.e., A contains a subalgebra B such that p|B : B ~→ A / I is an isomorphism.
Let R be a Noetherian integral domain with field of fractions K (for example, they can be Z, Q).
If the bilinear map A × A → A is reinterpreted as a linear map (i.e., morphism in the category of K-vector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms.
These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
If A and B are two algebras, and ρ : A → End(V) and τ : B → End(W) are two representations, then there is a (canonical) representation A ⊗ B → End(V ⊗ W) of the tensor product algebra A ⊗ B on the vector space V ⊗ W. However, there is no natural way of defining a tensor product of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions.
One might try to form a tensor product representation ρ : x ↦ σ(x) ⊗ τ(x) according to how it acts on the product vector space, so that However, such a map would not be linear, since one would have for k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ : A → A ⊗ A, and defining the tensor product representation as Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms.
Consider, for example, so that the action on the tensor product space is given by This map is clearly linear in x, and so it does not have the problem of the earlier definition.
However, it fails to preserve multiplication: But, in general, this does not equal This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel.
One example of a non-unital associative algebra is given by the set of all functions f : R → R whose limit as x nears infinity is zero.
Another example is the vector space of continuous periodic functions, together with the convolution product.