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  2. Zero divisor - Wikipedia

    en.wikipedia.org/wiki/Zero_divisor

    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.

  3. Zero-product property - Wikipedia

    en.wikipedia.org/wiki/Zero-product_property

    In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]

  4. Nilpotent - Wikipedia

    en.wikipedia.org/wiki/Nilpotent

    All nilpotent elements are zero divisors. An n × n {\displaystyle n\times n} matrix A {\displaystyle A} with entries from a field is nilpotent if and only if its characteristic polynomial is t n {\displaystyle t^{n}} .

  5. Idempotent (ring theory) - Wikipedia

    en.wikipedia.org/wiki/Idempotent_(ring_theory)

    Any non-trivial idempotent a is a zero divisor (because ab = 0 with neither a nor b being zero, where b = 1 − a). This shows that integral domains and division rings do not have such idempotents. Local rings also do not have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 0.

  6. Ring (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Ring_(mathematics)

    A left zero divisor of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0. [d] A right zero divisor is defined similarly. A nilpotent element is an element a such that a n = 0 for some n > 0. One example of a nilpotent element is a nilpotent matrix.

  7. Division by zero - Wikipedia

    en.wikipedia.org/wiki/Division_by_zero

    In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation, the general example can be written as a 0 {\displaystyle {\tfrac {a}{0}}} , where a {\displaystyle a} is the dividend (numerator).