Ad
related to: a is divisor of b means 0 in algebra 2 is known as quizlet
Search results
Results From The WOW.Com Content Network
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.
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).
But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression is undefined. In field theory, the expression is only shorthand for the formal expression ab −1, where b −1 is the multiplicative inverse of b.
1. In an integral domain R, [clarification needed] an element a is called a divisor of the element b (and we say a divides b) if there exists an element x in R with ax = b. 2. An element r of R is a left zero divisor if there exists a nonzero element x in R such that rx = 0 and a right zero divisor or if there exists a nonzero element y in R ...
If one interprets the definition of divisor literally, every a is a divisor of 0, since one can take x = 0. Because of this, it is traditional to abuse terminology by making an exception for zero divisors: one calls an element a in a commutative ring a zero divisor if there exists a nonzero x such that ax = 0. [2]
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.
A positive divisor of that is different from is called a proper divisor or an aliquot part of (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide n {\displaystyle n} but leaves a remainder is sometimes called an aliquant part of n . {\displaystyle n.}
In fact, every finite-dimensional real commutative division algebra is either 1- or 2-dimensional. This is known as Hopf's theorem, and was proved in 1940. The proof uses methods from topology. Although a later proof was found using algebraic geometry, no direct algebraic proof is known. The fundamental theorem of algebra is a corollary of Hopf ...