English Dictionary

zero divisor

名詞

(algebra, ring theory) An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

日本語の意味

環論において、環 R の元aであって、非零元x（x ≠ 0）と掛け合わせた際に、ax = 0 または xa = 0となるもの。これを日本語では「零因子」と呼びます。

When working in commutative algebra, it's important to determine whether an element is a zero divisor before attempting to invert it.

可換代数を扱う際、ある元を逆数にしようとする前に、それが非零の元と掛け合わせて零になるような環の元（零因子）であるかどうかを判断することが重要です。

意味（1）

(algebra, ring theory) An element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

An idempotent element e ne 1 of a ring is always a (two-sided) zero divisor, since e(1-e)=0=(1-e)e.

意味（2）

(algebra, ring theory) A nonzero element a of a ring R for which there exists some nonzero element x ∈ R such that either ax = 0 or xa = 0.

If R is an integral domain, that is, has no zero divisors, then R[x] also has no zero divisors.

zero divisors （ plural ）