group ring

noun

Given ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k.

Similar
group algebra
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Meanings

noungroup ring (plural group rings)

plural group rings

  1. algebraGiven ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k.

    Synonyms: group algebra

Synonyms & antonyms

nounGiven ring R with identity not equal to zero, and group G=g_1,g_2,...,g_n, the group ring RG has elements of the form a_1g_1+a_2g_2+...+a_ng_n (where a_i isin R) such that the sum of a_1g_1+a_2g_2+...+a_ng_n and b_1g_1+b_2g_2+...+b_ng_n is (a_1+b_1)g_1+(a_2+b_2)g_2+...+(a_n+b_n)g_n and the product is ∑ₖ₌₁ⁿ(∑_(g_ig_j=g_k)a_ib_j)g_k.

Similar
group algebra

Translations

Polish pierścień grupowy
  • pierścień grupowyTranslations
Portuguese anel de grupo
  • anel de grupoTranslations
Cite this entry
APA

group ring. (n.d.). In WatkinsX Dictionary. Retrieved November 3, 2026, from https://dictionary.watkinsx.com/word/group%20ring

MLA

“group ring.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/group%20ring. Accessed November 3, 2026.

Chicago

WatkinsX Dictionary, s.v. “group ring,” accessed November 3, 2026, https://dictionary.watkinsx.com/word/group%20ring.

Source: group ring on Wiktionary, CC BY-SA 4.0.