Simple algebra (universal algebra)
Today, Simple algebra (universal algebra) is a topic of great importance and interest for a wide spectrum of the population. As our society evolves and faces new challenges, the theme of Simple algebra (universal algebra) becomes a focal point for reflection and action. It is a topic that is present in all areas of life, from politics to pop culture, through technology and science. In this article, we will explore different aspects of Simple algebra (universal algebra) and its impact on our current society. We will address different perspectives, opinions and research findings to shed light on this topic that is so relevant in the contemporary world.
In universal algebra, an abstract algebra A is called simple if and only if 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 (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.[1]
See also
References
- ^ Lampe, W.A.; Taylor, W. (1982). "Simple algebras in varieties". Algebra Universalis. 14 (1): 36–43. doi:10.1007/BF02483905. S2CID 120637415. The original paper is Magari, R. (1969). "Una dimostrazione del fatto che ogni varietà ammette algebre semplici". Annalli dell'Università di Ferrara, Sez. VII (in Italian). 14 (1): 1–4. doi:10.1007/BF02896794. S2CID 115886103.