What is a k-algebra and what are examples?

[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

 

[mathwonk]

if by "k" you mean a field, then a k algebra is a (often commutative) ring A that contains k as a subring (contained in its center if the ring A is non commutative).so an algebra is in general just a relative version of a ring. in the commutative case, if k is a ring, a k algebra A is simply a ring homomorphism k-->A.

here k is not contained in A but by means of the homomorphism you can still multiply elements of A by elements of k.