Simple (abstract algebra)
In this article, we are going to delve into the fascinating world of Simple (abstract algebra). From its origins to its evolution over the years, we'll dive into everything related to Simple (abstract algebra). We will analyze its influence on various aspects of society, its impact on people's lives and its relevance in the current context. In addition, we will explore different perspectives and expert opinions on Simple (abstract algebra), with the aim of offering a global and complete vision of this exciting topic. Without a doubt, Simple (abstract algebra) is a topic that does not leave anyone indifferent and from which you can always learn something new.
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
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