Definition: Zero Divisor and Integral Domain

Let $$(R, + ,\cdot)$$ be a ring with $$1$$ as the identity of $$(R,\cdot)$$ and $$0$$ as the identity of $$(R,+)$$. Further, let $$a\in R$$.

• If $$a\neq 0$$ and there exists $$b\in R, b\neq 0$$ with
• $$a\cdot b=0$$, then we call $$a$$ a left zero divisor in $$R$$.
• $$b\cdot a=0$$, then we call $$a$$ a right zero divisor in $$R$$.
• we call $$a$$ a zero divisor in $$R$$, if $$a$$ is both, a left and a right zero divisor1.

A commutative ring $$R$$, which is not the zero ring and in which $$0$$ is the only zero divisor2 is called integral domain.

Branches: 1
Chapters: 2
Definitions: 3 4 5 6 7 8 9 10 11 12 13 14
Lemmas: 15 16 17 18
Parts: 19
Proofs: 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Propositions: 34 35 36 37 38 39 40 41 42
Theorems: 43 44

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References

Bibliography

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013

Footnotes

1. Please note that if $$R$$ is commutative, every left zero divisor is also a right zero divisor.

2. Please note that this is equivalent to $$a\cdot b=0\Leftrightarrow a=0 \vee b=0$$.