Simple (abstract algebra)
From Wikipedia, the free encyclopedia
In mathematics, the term simple is used to describe an algebraic structures which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is simple if the kernel of every homomorphism is either the whole structure or a single element. Some examples are:
- A group is a called a simple group if it does not contain a non-trivial proper normal subgroup.
- A ring is called a simple ring if it does not contain a non-trivial two sided ideal.
- A module is called a simple module if does not contain a non-trivial submodule.
- An algebra is called a simple algebra if does not contain a non-trivial two sided ideal.
The general pattern is that the structure admits no non-trivial congruence relations.
