specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and...

ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal...

commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is...

there is a distinction between a prime element and an irreducible element. An irreducible element x is an element such that if x = yz, then either y...

In mathematics, an element p of a partial order (P, ≤) is a meet prime element when p is the principal element of a principal prime ideal. Equivalently...

ring elements. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an...

The Prime Element is a jazz album by jazz drummer Elvin Jones, originally released in 1976 as part of the "Blue Note Re-issue Series". The pieces compiled...

which the zero ideal {0} is a prime ideal. An integral domain is a nonzero commutative ring for which every nonzero element is cancellable under multiplication...

irreducible element is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element z ∈...

numbers) shows. Every completely join-prime element of a lattice is compact.[citation needed] A poset in which every element is the supremum of the directed...

completely join prime. The dual notion is meet prime. Every join-prime element is also join irreducible, and every meet-prime element is also meet irreducible...

given below for the more general case. Note that an irreducible element of Z (a prime number) is still irreducible when viewed as constant polynomial...

theory) Prime 3-manifold, a 3-manifold that cannot be written as the connect sum of two nontrivial 3-manifolds Prime element, in algebra Prime form of...

ideal domain with a unique non-zero prime ideal. R is a principal ideal domain with a unique irreducible element (up to multiplication by units). R is...

an element f of R that is neither zero nor a unit is called irreducible if there are no non-units g and h with f = gh. One can show that every prime element...

activities to the local Gestapo office; thus the Block Warden was the prime element of the Nazi totalitarian state. This was helped by keeping files on...

an irreducible element is primal if and only if it is a prime element. Consequently, in a pre-Schreier domain, every irreducible is prime. In particular...

is a meet-prime element. Now the set-inverse of p −1(0) given by p −1(1) is a completely prime filter because p −1(0) is a principal prime ideal. It turns...

integers in the quadratic field, then an odd prime number p, not dividing d, is either a prime element in O d , {\displaystyle {\mathcal {O}}_{\sqrt...

first to use the term "element (στοιχεῖον, stoicheîon)" in reference to air, fire, earth, and water. The ancient Greek word for element, stoicheion (from stoicheo...

GF(q); this means that each non-zero element of GF(q) can be written as αi for some natural number i. If q is a prime number, the elements of GF(q) can be...

representation as a product of Fermi–Dirac primes without repetition. Each element of the sequence of Fermi–Dirac primes is the smallest number that does not...

decomposition of an element into irreducibles need not be unique; stated differently, an irreducible element is not necessarily a prime element. Important examples...

of powers of prime numbers. It is also known as the fundamental theorem of arithmetic. An element a {\displaystyle a} is a prime element if whenever a...

manufacturing. The manufacture of products or goods requires material as the prime element. In general, these materials are divided into two categories. These...

defined by the relation xa = σ(a)x ∀a ∈ R. prime 1.  An element x of an integral domain is a prime element if it is not zero and not a unit and whenever...

algorithm Fundamental theorem of arithmetic Irreducible element Prime element Prime number Prime ideal It is also called Euclid's first theorem although...

groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built. For any element g {\displaystyle...

operation is associative, it has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed...

that an incident "is in its very realism as much charged with that prime element of Radcliffian romance, the mysterious, as any that the ingenuity of...