Zero ring

In ring theory, a branch of mathematics, a zero ring is a ring (without unity) in which the product of any two elements is 0 (the additive identity element)^[1]. (Note: Some authors^[2] define a "zero ring" to be a ring with a single element, see trivial ring. These authors require their rings to have unity, hence all zero rings are trivial.) Another common name for zero rings is null ring; since the ring is not required to have unity, ideals can also be null rings, in which case they are referred to as null ideals.

Any abelian group can be turned into a zero ring by defining the product of any two elements to be 0. This proves that any abelian group is the additive group of some ring.

Any subgroup of the additive group of a zero ring is an ideal. It follows that the only zero rings that are simple are those whose additive groups are cyclic groups of prime order^[3].

References

1. ^ Bourbaki, Nicolas (1970) (in fr). Algèbre (Chapitres 1 à 3). Hermann. p. I-97.  with the denomination pseudo-anneau de carré nul
2. ^ For example Warner, Seth (1990). Modern Algebra. 1. Courier Dover. p. 188. ISBN 0-486-66341-8. 
3. ^ Zariski, Oscar; Samuel, Pierre (1958). Commutative Algebra. 1. Van Nostrand. p. 133. 

• Hall, Frederick Michael (1969). An Introduction to Abstract Algebra. 2. CUP Archive. p. 64. 

This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Ring theory Algebra stubs Wikimedia Foundation. 2010. Игры ⚽ Нужно сделать НИР? Null allomorph Null semigroup Look at other dictionaries: Ring (mathematics) — This article is about algebraic structures. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. Polynomials, represented here by curves, form a ring under addition and multiplication. In mathematics, a… …   Wikipedia Zero-product property — In the mathematical areas of algebra and analysis, the zero product property, also known as the zero product rule , is an abstract and explicit statement of the familiar property from elementary mathematics that if the product of two real numbers …   Wikipedia Zero-based numbering — is numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday circumstances. Under zero based numbering, the initial element is sometimes termed the zeroth element, rather than… …   Wikipedia Ring — /ring/, n. a male given name. * * * I Circular band of gold, silver, or other precious or decorative material usually worn on the finger, but sometimes on the toes, the ears, or the nose. The earliest examples were found in the tombs of ancient… …   Universalium Ring (Bruxelles) — Ring de Bruxelles Ring de Bruxelles …   Wikipédia en Français Ring 0 — Ring de Bruxelles Ring de Bruxelles …   Wikipédia en Français Ring Bruxelles — Ring de Bruxelles Ring de Bruxelles …   Wikipédia en Français Ring belge R0 — Ring de Bruxelles Ring de Bruxelles …   Wikipédia en Français Ring de Bruxelles — Longueur 75 km (dont 62 km autoroutiers[1] …   Wikipédia en Français Ring de bruxelles — Ring de Bruxelles …   Wikipédia en Français 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”