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Unit (ring theory) — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Unit_(ring_theory)
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-15T16:01:19.243Z
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Unit (ring theory) — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Unit_(ring_theory) (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_2da10aba05aa (Metraipolis archive dc_2da10aba05aa, sha256 09cc2afd57e152b69830c6bf882218481941153da1afe233f9be4668eab659d2)

BibTeX

@misc{dc2da10aba05aa,
  title = {Unit (ring theory) — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Unit_(ring_theory)},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_2da10aba05aa; sha256 09cc2afd57e152b69830c6bf882218481941153da1afe233f9be4668eab659d2},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Unit_(ring_theory). Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that v u = u v = 1 , {\displaystyle vu=uv=1,} where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group R× under multiplication, called the group of units or unit group of R. Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit). Less commonly, the term unit is sometimes used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, 1 is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «number theory»).
Fonte: Wikipedia in inglese, voce «Unit (ring theory)» — https://en.wikipedia.org/wiki/Unit_(ring_theory)
Wikidata Q118084 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q118084
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Unit_(ring_theory)?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1309407347, consultata il 2026-10-15T15:25:41.706Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 4d6a4dd4e6afd7d1898c3176f29a3c0837584e3cc01e8cf68760d0e757dee193