In
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of stu ...
, an abstract algebra ''A'' is called ''simple''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
it has no nontrivial
congruence relations, or equivalently, if every homomorphism with domain ''A'' is either
injective or constant.
As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the notion from ring theory: a ring is simple in the sense that it has no nontrivial ideals if and only if it is simple in the sense of universal algebra. The same remark applies with respect to groups and normal subgroups; hence the universal notion is also a generalization of a
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
(it is a matter of convention whether a one-element algebra should be or should not be considered simple, hence only in this special case the notions might not match).
A theorem by
Roberto Magari in 1969 asserts that every variety contains a simple algebra.
[ The original paper is ]
See also
*
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
*
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a sim ...
*
central simple algebra
References
{{Reflist
Algebras
Ring theory