Zero divisor

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In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0,[1] or equivalently if the map from R to R that sends x to ax is not injective.[a] Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. 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.[2] An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). 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,[3] 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.

Examples[edit]

  • In the ring , the residue class is a zero divisor since .
  • The only zero divisor of the ring of integers is .
  • A nilpotent element of a nonzero ring is always a two-sided zero divisor.
  • An idempotent element of a ring is always a two-sided zero divisor, since .
  • The ring of matrices over a field has nonzero zero divisors if . Examples of zero divisors in the ring of matrices (over any nonzero ring) are shown here:
    .
  • A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in with each nonzero, , so is a zero divisor.
  • Let be a field and be a group. Suppose that has an element of finite order . Then in the group ring one has , with neither factor being zero, so is a nonzero zero divisor in .

One-sided zero-divisor[edit]

  • Consider the ring of (formal) matrices with and . Then and . If , then is a left zero divisor if and only if is even, since , and it is a right zero divisor if and only if is even for similar reasons. If either of is , 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 be the set of all sequences of integers . Take for the ring all additive maps from to , with pointwise addition and composition as the ring operations. (That is, our ring is , the endomorphism ring of the additive group .) Three examples of elements of this ring are the right shift , the left shift , and the projection map onto the first factor . All three of these additive maps are not zero, and the composites and are both zero, so is a left zero divisor and is a right zero divisor in the ring of additive maps from to . However, is not a right zero divisor and is not a left zero divisor: the composite is the identity. is a two-sided zero-divisor since , while is not in any direction.

Non-examples[edit]

Properties[edit]

  • In the ring of n-by-n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n-by-n 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 a is invertible and ax = 0 for some nonzero x, then 0 = a−10 = a−1ax = x, a contradiction.
  • An element is cancellable on the side on which it is regular. That is, if a is a left regular, ax = ay implies that x = y, and similarly for right regular.

Zero as a zero divisor[edit]

There is no need for a separate convention for the case a = 0, because the definition applies also in this case:

  • If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because any nonzero element x satisfies 0x = 0 = x0.
  • If R is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0.

Some references include or exclude 0 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 R, the set of non-zero-divisors is a multiplicative set in R. (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 R, the set of zero divisors is the union of the associated prime ideals of R.

Zero divisor on a module[edit]

Let R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is M-regular if the "multiplication by a" map is injective, and that a is a zero divisor on M otherwise.[4] The set of M-regular elements is a multiplicative set in R.[4]

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

See also[edit]

Notes[edit]

  1. ^ Since the map is not injective, we have ax = ay, in which x differs from y, and thus a(xy) = 0.

References[edit]

  1. ^ N. Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag, p. 98
  2. ^ Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
  3. ^ Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 15.
  4. ^ a b Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing Company, Inc., p. 12

Further reading[edit]