Regular element (ring theory)

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element  that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same.

An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable).
An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non- zero ring with no nontrivial zero divisors is called a domain.

Examples

* In the ring $\mathbb/4\mathbb$, the residue class $\overline$ is a zero divisor since $\overline \times \overline=\overline=\overline$.
* The only zero divisor of the ring $\mathbb$ of integers is $0$.
* A nilpotent element of a nonzero ring is always a two-sided zero divisor.
* An idempotent element $e\ne 1$ of a ring is always a two-sided zero divisor, since $e(1-e)=0=(1-e)e$.
* The ring of ''n'' × ''n'' matrices over a field has nonzero zero divisors if ''n'' ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here:
$\begin1&1\\2&2\end\begin1&1\\-1&-1\end=\begin-2&1\\-2&1\end\begin1&1\\2&2\end=\begin0&0\\0&0\end ,$ $\begin1&0\\0&0\end\begin0&0\\0&1\end =\begin0&0\\0&1\end\begin1&0\\0&0\end =\begin0&0\\0&0\end.$
* A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in $R_1 \times R_2$ with each $R_i$ nonzero, $(1,0)(0,1) = (0,0)$, so $(1,0)$ is a zero divisor.
* Let $K$ be a field and $G$ be a group. Suppose that $G$ has an element $g$ of finite order $n > 1$. Then in the group ring K /math> one has $(1-g)(1+g+ \cdots +g^)=1-g^=0$, with neither factor being zero, so $1-g$ is a nonzero zero divisor in K /math>.

One-sided zero-divisor

* Consider the ring of (formal) matrices $\beginx&y\\0&z\end$ with $x,z\in\mathbb$ and $y\in\mathbb/2\mathbb$. Then $\beginx&y\\0&z\end\begina&b\\0&c\end=\beginxa&xb+yc\\0&zc\end$ and $\begina&b\\0&c\end\beginx&y\\0&z\end=\beginxa&ya+zb\\0&zc\end$. If $x\ne0\ne z$, then $\beginx&y\\0&z\end$ is a left zero divisor if and only if $x$ is even, since $\beginx&y\\0&z\end\begin0&1\\0&0\end=\begin0&x\\0&0\end$, and it is a right zero divisor if and only if $z$ is even for similar reasons. If either of $x,z$ is $0$, then it is a two-sided zero-divisor.
*Here is another example of a ring with an element that is a zero divisor on one side only. Let $S$ be the set of all sequences of integers $(a_1,a_2,a_3,...)$. Take for the ring all additive maps from $S$ to $S$, with pointwise addition and composition as the ring operations. (That is, our ring is $\mathrm(S)$, the ''endomorphism ring'' of the additive group $S$.) Three examples of elements of this ring are the right shift $R(a_1,a_2,a_3,...)=(0,a_1,a_2,...)$, the left shift $L(a_1,a_2,a_3,...)=(a_2,a_3,a_4,...)$, and the projection map onto the first factor $P(a_1,a_2,a_3,...)=(a_1,0,0,...)$. All three of these additive maps are not zero, and the composites $LP$ and $PR$ are both zero, so $L$ is a left zero divisor and $R$ is a right zero divisor in the ring of additive maps from $S$ to $S$. However, $L$ is not a right zero divisor and $R$ is not a left zero divisor: the composite $LR$ is the identity. $RL$ is a two-sided zero-divisor since $RLP=0=PRL$, while $LR=1$ is not in any direction.

Non-examples

* The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field.
* More generally, a division ring has no nonzero zero divisors.
* A non-zero commutative ring whose only zero divisor is 0 is called an integral domain.

Properties

* In the ring of  ×  matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of  ×  matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero.
* Left or right zero divisors can never be units, because if is invertible and for some nonzero , then , a contradiction.
* An element is cancellable on the side on which it is regular. That is, if is a left regular, implies that , and similarly for right regular.

Zero as a zero divisor

There is no need for a separate convention for the case , because the definition applies also in this case:
* If is a ring other than the zero ring, then is a (two-sided) zero divisor, because any nonzero element satisfies .
* If is the zero ring, in which , then is not a zero divisor, because there is no ''nonzero'' element that when multiplied by yields .

Some references include or exclude as a zero divisor in ''all'' rings by convention, but they then suffer from having to introduce exceptions in statements such as the following:
* In a commutative ring , the set of non-zero-divisors is a multiplicative set in . (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
* In a commutative noetherian ring , the set of zero divisors is the union of the associated prime ideals of .

Zero divisor on a module

Let be a commutative ring, let be an - module, and let be an element of . One says that is -regular if the "multiplication by " map $M \,\stackrel\to\, M$ is injective, and that is a zero divisor on otherwise. The set of -regular elements is a multiplicative set in .

Specializing the definitions of "-regular" and "zero divisor on " to the case recovers the definitions of "regular" and "zero divisor" given earlier in this article.

See also

* Zero-product property
* Glossary of commutative algebra (Exact zero divisor)
* Zero-divisor graph
* Sedenions, which have zero divisors

Notes

References

Further reading

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* {{MathWorld , title=Zero Divisor , urlname=ZeroDivisor

Abstract algebra
Ring theory
0 (number)
Sedenions