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- Zero divisor

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 is called **left regular** or **left cancellable**. Similarly, an element of a ring that is not a right zero divisor is called **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 nonzero ring with no nontrivial zero divisors is called a domain.

- In the ring , the residue class
Z

*/*4Z

*\overline{*2*}*

*\overline{*2*}* x *\overline{*2*}*=*\overline{*4*}*=*\overline{*0*}*

- The only zero divisor of the ring

Z

0

- A nilpotent element of a nonzero ring is always a two-sided zero divisor.

*e\ne*1

*e(*1-*e)*=0=*(*1-*e)e*

- The ring of matrices over a field has nonzero zero divisors if
*n*x*n*

*n**\geq*2

2 x 2

*\begin{pmatrix}*1*&*1*\\*2*&*2*\end{pmatrix}\begin{pmatrix}*1*&*1*\\*-1*&*-1*\end{pmatrix}*=*\begin{pmatrix}*-2*&*1*\\*-2*&*1*\end{pmatrix}\begin{pmatrix}*1*&*1*\\*2*&*2*\end{pmatrix}*=*\begin{pmatrix}*0*&*0*\\*0*&*0*\end{pmatrix}**,*

*\begin{pmatrix}*1*&*0*\\*0*&*0*\end{pmatrix}\begin{pmatrix}*0*&*0*\\*0*&*1*\end{pmatrix}
*=*\begin{pmatrix}*0*&*0*\\*0*&*1*\end{pmatrix}\begin{pmatrix}*1*&*0*\\*0*&*0*\end{pmatrix}
*=*\begin{pmatrix}*0*&*0*\\*0*&*0*\end{pmatrix}*

- A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in

*R*_{1} x *R*_{2}

*R*_{i}

*(*1*,*0*)(*0*,*1*)*=*(*0*,*0*)*

*(*1*,*0*)*

- Let

*K*

*G*

*G*

*g*

*n>*1

*K[G]*

*(*1-*g)(*1+*g*+ … +*g*^{n-1}*)*=1-*g*^{n}=0

1-*g*

*K[G]*

- Consider the ring of (formal) matrices

*\begin{pmatrix}x&y\\*0*&z\end{pmatrix}*

*x,z\inZ*

*y\inZ/*2Z

*\begin{pmatrix}x&y\\*0*&z\end{pmatrix}\begin{pmatrix}a&b\\*0*&c\end{pmatrix}*=*\begin{pmatrix}xa&xb*+*yc\\*0*&zc\end{pmatrix}*

*\begin{pmatrix}a&b\\*0*&c\end{pmatrix}\begin{pmatrix}x&y\\*0*&z\end{pmatrix}*=*\begin{pmatrix}xa&ya*+*zb\\*0*&zc\end{pmatrix}*

*x\ne*0*\ne**z*

*\begin{pmatrix}x&y\\*0*&z\end{pmatrix}*

*x*

*\begin{pmatrix}x&y\\*0*&z\end{pmatrix}\begin{pmatrix}*0*&*1*\\*0*&*0*\end{pmatrix}*=*\begin{pmatrix}*0*&x\\*0*&*0*\end{pmatrix}*

*z*

*x,z*

0

- Here is another example of a ring with an element that is a zero divisor on one side only. Let

*S*

*(a*_{1,a}_{2,a}_{3,...)}

*S*

*S*

End*(S)*

*S*

*R(a*_{1,a}_{2,a}_{3,...)=(0,a}_{1,a}_{2,...)}

*L(a*_{1,a}_{2,a}_{3,...)=(a}_{2,a}_{3,a}_{4,...)}

*P(a*_{1,a}_{2,a}_{3,...)=(a}_{1,0,0,...)}

*LP*

*PR*

*L*

*R*

*S*

*S*

*L*

*R*

*LR*

*RL*

*RLP*=0=*PRL*

*LR*=1

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

- In the ring of -by- matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of -by- 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.

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 .

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{a}\to**M*

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.

- Zero-product property
- Glossary of commutative algebra (Exact zero divisor)
- Zero-divisor graph