In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the map...

in combinatorial commutative algebra, a zero-divisor graph is an undirected graph representing the zero divisors of a commutative ring. It has elements...

divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors...

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...

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...

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). A...

z} of a Banach algebra A {\displaystyle A} is called a topological divisor of zero if there exists a sequence x 1 , x 2 , x 3 , . . . {\displaystyle x_{1}...

In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation,...

trivial group {0}. The element 0 in the zero ring is not a zero divisor. The only ideal in the zero ring is the zero ideal {0}, which is also the unit ideal...

the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive...

additive identity among those tensors. Null semigroup Zero divisor Zero object Zero of a function Zero — non-mathematical uses Nair, M. Thamban; Singh, Arindama...

set up divisor (13) for second division loop (MEMORY LAYOUT: zero copy dividend divisor remainder quotient zero zero) >-[>+>>] Reduce divisor; Normal...

not a division algebra because they have zero divisors: two nonzero sedenions can be multiplied to obtain zero, for example ( e 3 + e 10 ) ( e 6 − e 15...

nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements x, y such that xy = 0. A square matrix has...

algebra and f a homogeneous element of degree d in A which is not a zero divisor. Then we have H S A / ( f ) ( t ) = ( 1 − t d ) H S A ( t ) . {\displaystyle...

commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring...

is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple...

torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the...

In mathematics, a divisor of an integer n , {\displaystyle n,} also called a factor of n , {\displaystyle n,} is an integer m {\displaystyle m} that may...

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. A right zero divisor is defined...

torsion-free group. Kaplansky's zero divisor conjecture states: The group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Two...

The Zero Divisor Theorem. If M ≠ 0 {\displaystyle M\neq 0} has finite projective dimension and r ∈ R {\displaystyle r\in R} is not a zero divisor on M...

+ (xy) as zero-divisors, but no non-zero nilpotent elements. As another example, the ring Z × Z contains (1, 0) and (0, 1) as zero-divisors, but contains...

adjoin a new zero 0 ′ {\displaystyle 0'} to the underlying set and thus obtain such a zerosumfree semiring that also lacks zero divisors. In particular...

y {\displaystyle y} is in I {\displaystyle I} ; equivalently, every zero-divisor in the quotient R / I {\displaystyle R/I} is nilpotent. The radical of...

module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module...

{\displaystyle s\in S,} and 0 ≠ a ∈ R {\displaystyle 0\neq a\in R} is a zero divisor with a s = 0. {\displaystyle as=0.} Then a 1 {\displaystyle {\tfrac {a}{1}}}...

In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a...

form of Euclid's algorithm. exact zero divisor A zero divisor x {\displaystyle x} is said to be an exact zero divisor if its annihilator, Ann R ⁡ ( x )...

then R[G] always has zero divisors. For example, consider an element g of G of order |g| = m > 1. Then 1 − g is a zero divisor: ( 1 − g ) ( 1 + g + ⋯...