Simple (abstract algebra)

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Published on March 14, 2022 1:42:19 AM ● Video Link: https://www.youtube.com/watch?v=Uug1oIs7Z_s

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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.

Source: https://en.wikipedia.org/wiki/Simple_(abstract_algebra)
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