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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.
When R is commutative, the notions of left divisor, right divisor, and two-sided divisor coincide, so one says simply that a is a divisor of b, or that b is a multiple of a, and one writes . Elements a and b of an integral domain are associates if both a ∣ b {\displaystyle a\mid b} and b ∣ a {\displaystyle b\mid a} .
The graph of all zero divisors is non-empty for every ring that is not an integral domain. It remains connected, has diameter at most three, [3] and (if it contains a cycle) has girth at most four. [4] [5] The zero-divisor graph of a ring that is not an integral domain is finite if and only if the ring is finite. [3]
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] ( Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor).
Let R be an effective commutative ring.. There is an algorithm for testing if an element a is a zero divisor: this amounts to solving the linear equation ax = 0.; There is an algorithm for testing if an element a is a unit, and if it is, computing its inverse: this amounts to solving the linear equation ax = 1.
The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide. [1] [6] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0. The proof of the last equality is found here.) The zero ring is commutative. The element 0 in the zero ring is a unit, serving as its own multiplicative ...
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A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero zero divisors. Whenever A is an associative unital algebra over the field F and S is a simple module over A , then the endomorphism ring of S is a division algebra over F ; every associative division algebra over F arises in ...