ok, you know how fields can be considered "two groups in one" (one is the additive group, and one is the multiplicative group of non-zero elements)? well a k-algebra is a similar idea of "two structures in one".
on one hand, we have that a k-algebra is a vector space over k. this is the same thing as a k-module (which is FREE over any basis).
on the other hand, we have that a k-algebra forms a ring (usually associative and with unity). the ring structure and the vector space structure have to be COMPATIBLE.
this means: the ring multiplication is bilinear.
another way to look at this is that you have a ring, A, together with a ring-homomorphism η:k-->Z(A) (the center of A).
if A is not a trivial ring (just 0), then η is injective (because k is a field, and field-homomorphisms are always monomorphisms), so normally n(k) is identified with k.
this allows us to define a vector space on A by:
vector addition is just the ring addition (for any ring A, (A,+) is an abelian group).
scalar multiplication is defined like so: for c in k, and a in A:
ca = η(c)a (where the RHS is the ring multiplication of A).
some typical examples:
let k be a field, and let E be any field containing k as a subfield. define for c in k, and a in E: ca to be the product in E. for example, the complex numbers C are an R-algebra. this is a 2-dimensional R-algebra.
let k be a field, and let k[x] be the ring of polynomials over k. then k[x] is a k-algebra. this is an infinite-dimensional k-algebra.
let k be a field, and let V be any vector space over k. then Hom(V,V) = End(V), the set of all k-linear mappings from V to V, is a k-algebra. if V is finite-dimensional, of dimension n, then End(V) has dimension n2. for the finite-dimensional case, this can be identified with Matnxn(k), the set of all nxn matrices with entries in k.
let G be any group, and let k be any field. the then group-algebra k[G], consisting of formal k-linear combinations of elements of G (together with a "polynomial-like" multiplication) is a k-algebra.
the fact that k-algebras are vector spaces over k, lets us use the tools of linear algebra to investigate them. the fact the k-algebras are also rings, lets us use ring-concepts in investigating them.
for example, for any k-algebra, we can speak of its group of units. for the k-algebra k[x], this is just k. for the k-algebra Endk(V), this is GLk(V), the general linear group of V. again, if V is finite-dimensional, this corresponds to the INVERTIBLE nxn matrices.
one can also form "function algebras". for example, one has the R-algebra C[a,b], consisting of continuous functions f:[a,b]--->R, where the ring-structure is inherited from R:
(f+g)(x) = f(x) + g(x)
(fg)(x) = f(x)g(x)
(cf)(x) = c(f(x)) <--this is the "scalar multiplication".
this example can easily be generalized to functions f:S-->k, where S is any set, and k is the field k. often we are interested in some sub-ring of kS (continuous, differentiable, linear, etc.) and often S has additional properties (such as a topology, or is a vector space itself).
historically, k-algebras (and not specific instances of them) are relatively recent, being studied in their own right only since around the beginning of the 20th century. some of the development of k-algebras probably came about as an attempt to realize various structures as matrix algebras, for example it is well-known that the complex numbers can be realized as a sub-algebra of Mat2x2(R) as:
in a similar vein, the quaternions form a 4-dimensional algebra over R, and a 2-dimensional algebra over C, a 2x2 complex matrix that describes a quaternion is:
where A* = the complex conjugate of A. each entry can be viewed as a 2x2 "block matrix", yielding a 4x4 real matrix.
a nifty feature of End(V), for finite-dimensional V: we have a monoid-homomorphism det:End(V)-->k, to the multiplicative monoid (k,*), which preserves units: T is in U(End(V)) (that is: GL(V)) iff det(T) is in U(k) = k* (that is: if det(T) ≠ 0).
in general, we can investigate the sub-structure of a k-algebra A, by considering the ideals of A as a ring. such ideals are automatically subspaces of A since:
u in J and v in J means u+v is in J (ideals are closed under addition)
u in J and c in k means cu is in J (here we are implicitly considering c in A via the monomorphism η).
0 is in any ideal J of A.
note that in the algebra k[x], the ideal generated by x is considerably bigger than the subspace generated by x. so in k-algebras, you sometimes have to be careful specifying "how you're decomposing it". the "dual nature" of k-algebras leads to a rich and varied theory. one of the things k-algebras are useful for is "representation theory". a representation of a k-algebra A consists of two things:
1. a vector space V over k
2. an action on V by A via endomorphisms (equivalently: an algebra homomorphism A --> Endk(V)).
concretely (when dimension V = n), this lets us think of elements of A as nxn matrices, so that instead of doing "abstract algebra" in A, we can do "concrete (matrix) arithmetic" in Matnxn(k). this, in turn, is equivalent to turning V into an A-module (you may already be familiar with regarding V as a k[x]-module, by picking a linear transformation T, and setting p(x).v = p(T)(v), or regarding V as a k[G]-module via a homomorphism φ:G-->GL(V) and setting:
(a1e + a2g1 +...+ angn-1).v = a1v + a2φ(g1)(v) +...+ anφ(gn-1)(v) ).