English Dictionary

ring of fractions

名詞

(algebra) A ring whose elements are fractions whose numerators belong to a given commutative unital ring and whose denominators belong to a multiplicatively closed unital subset D of that given ring. Addition and multiplication of such fractions is defined just as for a field of fractions. A pair of fractions a/b and c/d are deemed equivalent if there is a member x of D such that x(ad-bc)=0.

日本語の意味

(代数学) ある可換単位元付き環と、その環内の乗法的に閉じた単位集合Dに対して、分子がその環の元、分母がDの元である分数たちを要素とする環。これらの分数の加法および乗法は、分数体の場合と同様に定義され、例えばa/bとc/dは、Dの元xが存在してx(ad - bc) = 0となるときに同値とみなされる。

To construct the ring of fractions, one inverts every element of the multiplicatively closed subset D of a commutative ring R and forms equivalence classes of pairs (a, d) representing fractions a/d.

分数環を構成するには、可換環Rの乗法閉部分集合Dの各元を逆元にして、分数a/dを表す対(a, d)の同値類を形成する。

意味（1）

(algebra) A ring whose elements are fractions whose numerators belong to a given commutative unital ring and whose denominators belong to a multiplicatively closed unital subset D of that given ring. Addition and multiplication of such fractions is defined just as for a field of fractions. A pair of fractions a/b and c/d are deemed equivalent if there is a member x of D such that x(ad-bc)=0.

rings of fractions （ plural ）