Simple (abstract algebra)

In mathematics, the term simple is used to describe an algebraic structure 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 called a simple group if it does not contain a nontrivial proper normal subgroup.
• A ring is called a simple ring if it does not contain a nontrivial two sided ideal.
• A module is called a simple module if it does not contain a nontrivial submodule.
• An algebra is called a simple algebra if it does not contain a nontrivial two sided ideal.

The general pattern is that the structure admits no non-trivial congruence relations.

The term is used differently in semigroup theory. A semigroup is said to be simple if it has no nontrivial ideals, or equivalently, if Green's relation J is the universal relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may have nontrivial congruences. A semigroup with no nontrivial congruences is called congruence simple.

See also[]

• semisimple
• simple universal algebra

Index of articles associated with the same name

This article includes a list of related items that share the same name (or similar names).
If an incorrectly led you here, you may wish to change the link to point directly to the intended article.