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 Group (mathematics), groups as ...
, 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 bi ...
it has no nontrivial
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
s, or equivalently, if every homomorphism with domain ''A'' is either
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
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 da ...
(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 da ...
*
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 simple ...
*
central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simp ...
References
{{Reflist
Algebras
Ring theory