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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 the factor ring /, the equivalence class of 3 is nilpotent because 3 2 is congruent to 0 modulo 9. Assume that two elements a {\displaystyle a} and b {\displaystyle b} in a ring R {\displaystyle R} satisfy a b = 0 {\displaystyle ab=0} .
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.
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 ).
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
Take each digit of the number (371) in reverse order (173), multiplying them successively by the digits 1, 3, 2, 6, 4, 5, repeating with this sequence of multipliers as long as necessary (1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, ...), and adding the products (1×1 + 7×3 + 3×2 = 1 + 21 + 6 = 28). The original number is divisible by 7 if and only if ...
Let X be the subvariety of the four-dimensional affine plane, with coordinates x,y,z,w, generated by y 2-x 3 and x 4 +xz 2-w 3. The canonical desingularization of the ideal with these generators would blow up the center C 0 given by x=y=z=w=0. The transform of the ideal in the x-chart if generated by x-y 2 and y 2 (y 2 +z 2-w 3).
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.