Ads
related to: zero divisors calculator multiplication worksheet pdf printable forms sheets
Search results
Results From The WOW.Com Content Network
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]
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.
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.
As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition. [ 4 ] [ 5 ] More generally, in view of the above definitions, an operator Q {\displaystyle Q} is nilpotent if there is n ∈ N {\displaystyle n\in \mathbb {N} } such that Q n = 0 {\displaystyle Q^{n}=0} (the zero function ).
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.
Considered with the operations of component-wise addition, and multiplication according to the quaternion group, this collection forms a 4-dimensional algebra over the complex numbers C. The algebra of biquaternions is associative, but not commutative. A biquaternion is either a unit or a zero divisor.
Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain if and only if it is reduced and its spectrum Spec R is an irreducible topological space. The first property is often considered to encode some infinitesimal information, whereas the second one is more geometric.
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]