Last Updated：2025/11/25
Sentence
抽象代数学において、群環RGは係数a_iがRに属する形式和a_1g_1 + a_2g_2 + ... + a_ng_nから成り、加法は(a_1g_1+...+a_ng_n)+(b_1g_1+...+b_ng_n) = (a_1+b_1)g_1+...+(a_n+b_n)g_nであり、乗法は∑_{k=1}^n (∑_{g_i g_j = g_k} a_i b_j) g_kによって与えられます。
Quizzes for review
In abstract algebra, the group ring RG consists of formal sums a_1g_1 + a_2g_2 + ... + a_ng_n with coefficients a_i in R, where (a_1g_1+...+a_ng_n)+(b_1g_1+...+b_ng_n) = (a_1+b_1)g_1+...+(a_n+b_n)g_n and the product is given by ∑{k=1}n (∑{g_i g_j = g_k} a_i b_j) g_k.
See correct answer
In abstract algebra, the group ring RG consists of formal sums a_1g_1 + a_2g_2 + ... + a_ng_n with coefficients a_i in R, where (a_1g_1+...+a_ng_n)+(b_1g_1+...+b_ng_n) = (a_1+b_1)g_1+...+(a_n+b_n)g_n and the product is given by ∑{k=1}n (∑{g_i g_j = g_k} a_i b_j) g_k.
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Related words
group ring
Noun
(algebra)
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.
Japanese Meaning
群環：環 R（単位元を含みゼロと異なる環）と群 G の各元を用いて、形式線形結合 a₁g₁ + a₂g₂ + … + aₙgₙ （aᵢ ∈ R）の形で構成される環。加法および乗法はそれぞれ対応する係数の加算と、群の積に従った分配法則によって定義される。
Related Words
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