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In mathematics, an element of a ring is called nilpotent if there exists some positive integer, called the index (or sometimes the degree), such that =.. The term, along with its sister idempotent, was introduced by Benjamin Peirce in the context of his work on the classification of algebras.
More generally, every integral domain is a reduced ring since a nilpotent element is a fortiori a zero-divisor. On the other hand, not every reduced ring is an integral domain; for example, the ring Z [ x , y ]/( xy ) contains x + ( xy ) and y + ( xy ) as zero-divisors, but no non-zero nilpotent elements.
In linear algebra, a nilpotent matrix is a square matrix N such that = for some positive integer.The smallest such is called the index of , [1] sometimes the degree of .. More generally, a nilpotent transformation is a linear transformation of a vector space such that = for some positive integer (and thus, = for all ).
A simple and sufficient test for the absence of a dependence is the greatest common divisor (GCD) test. It is based on the observation that if a loop carried dependency exists between X[a*i + b] and X[c*i + d] (where X is the array; a, b, c and d are integers, and i is the loop variable), then GCD (c, a) must divide (d – b).
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero.
Divide the first term of the dividend by the highest term of the divisor (x 3 ÷ x = x 2). Place the result below the bar. x 3 has been divided leaving no remainder, and can therefore be marked as used by crossing it out. The result x 2 is then multiplied by the second term in the divisor −3 = −3x 2.
Printable Crossword Puzzle: September 2017 We've used the names of Snow White's diminutive friends as clues in this crossword. How they are defined is up to you to determine. Here's a tip: If you ...
In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or y n is also an element of Q, for some n > 0. For example, in the ring of integers Z, (p n) is a primary ideal if p is a prime number.