In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply called a zero divisor. If multiplication in the ring is commutative, then the left and right zero divisors are the same. A nonzero element of a ring that is neither a left nor right zero divisor is called regular.

Examples

• The ring Z of integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  s has no zero divisors, but in the ring Z × Z with componentwise addition and multiplication, (0,1)·(1,0) = (0,0), so both (0,1) and (1,0) are zero divisors.
• An example of a zero divisor in the ring of 2-by-2 matrices
  Matrix (mathematics)
  In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
  
   is the matrix
  because for instance
  
  • More generally in the ring of n-by-n matrices over some field
    Field (mathematics)
    In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
    
    , the left and right zero divisors coincide; they are precisely the nonzero singular matrices. In the ring of n-by-n matrices over some integral domain, the zero divisors are precisely the nonzero matrices with determinant
    Determinant
    In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
    
     zero
    0 (number)
    0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
    
    .
  
  
  • Here is an 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 (a₁, a₂, a3...). Take for the ring all additive maps
    Additive function
    In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...
    
     from S to S, with pointwise addition and composition as the ring operations. (That is, our ring is End(S), the endomorphism
    Endomorphism
    In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
    
    s of the additive group S.) Three examples of elements of this ring are the right shift R(a₁, a₂, a3,...) = (0, a₁, a₂,...), the left shift L(a₁, a₂, a3,... ) = (a₂, a₃,...), and a third additive map T(a₁, a₂, a3,... ) = (a₁, 0, 0, ... ). All three of these additive maps
    Additive function
    In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...
    
     are not zero, and the composites LT and TR 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, so if some additive map
    Additive function
    In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...
    
     f from S to S satisfies fL= 0 then composing both sides of this equation on the right with R shows (fL)R = f(LR) = f1 = f has to be 0, and similarly if some f satisfies Rf = 0 then composing both sides on the left with L shows f is 0.
  
  
  Continuing with this example, note that while RL is a left zero divisor ((RL)T = R(LT) is 0 because LT is), LR is not a
  zero divisor on either side because it is the identity.
  
  Concretely, we can interpret additive maps from S to S as countably infinite matrices. The matrix
  realizes L explicitly (just apply the matrix to a vector and see the effect is exactly a left shift)
  and the transpose B = A^T
  Transpose
  In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
  
   realizes the right shift on S. That AB is the identity matrix
  Identity matrix
  In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
  
   is the same as saying LR is the identity.
  In particular, as matrices A is a left zero divisor but not a right zero divisor.
  

  Properties

  Left or right zero divisors can never be unit
  Unit (ring theory)
  In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
  
  s, because if a is invertible and ab = 0, then 0 = a⁻¹0 = a⁻¹ab = b.
  
  Every nonzero idempotent element a ≠ 1 is a zero divisor, since a² = a implies a(a − 1) = (a − 1)a = 0. Nonzero nilpotent
  Nilpotent
  In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
  
   ring elements are also trivially zero divisors.
  
  A commutative ring
  Commutative ring
  In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
  
   with 0 ≠ 1 and without zero divisors is called an integral domain.
  
  Zero divisors occur in the quotient ring
  Quotient ring
  In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
  
   Z/nZ if and only if n is composite
  Composite number
  A composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
  
  . When n is prime
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
  , there are no zero divisors and this ring
  Ring (mathematics)
  In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
  
   is, in fact, a field
  Field (mathematics)
  In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
  
  , as every nonzero element is a unit
  Unit (ring theory)
  In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
  
  .
  
  Zero divisors also occur in the sedenion
  Sedenion
  In abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions...
  
  s, or 16-dimension
  Dimension
  In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
  
  al hypercomplex number
  Hypercomplex number
  In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
  
  s under the Cayley–Dickson construction.
  
  The set of zero divisors is the union of the associated prime ideals of the ring.
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