K-Algebra - Meaning and background of the concept

[Math Amateur]

I am trying to understand the concept of a k-algebra without much success.

Can someone please give me a clear explanation of the background, definition and use of the concept.

Also I would be extremely grateful of some examples of k-algebras

Peter

 

[Turgul]

The basic ideal of a $k$-algebra is quite simple. A $k$-algebra is merely a $k$ vector space which also happens to be a ring.

The prototypical example is the polynomial ring $k[x_1, \ldots ,x_n]$. A noncommutative $k$-algebra would be $M_n(k)$, the $n \times n$ matrices with entries in $k$. Both of these objects are rings, you can add or multiply two polynomials together, as with two matrices, but you can also multiply everything by an element of $k$. More concretely, $\mathbb{C}$ is an $\mathbb{R}$-algebra.

Some convenient properties of a $k$-algebra $R$ is that any quotient ring of $R$ must still be a $k$-algebra and any $R$-module must be a $k$ vector space. Furthermore, one has a rigid notion of dimension of a $k$-algebra (just the dimension of the underlying vector space) which can often be helpful. Unfortunately however, in many cases this dimension is infinite (even in $k[x]$ the dimension is infinite as a basis is the set of monomials $x^n$).