ascending chain condition वाक्य
उदाहरण वाक्य
मोबाइल
- The Artin Wedderburn theorem generalises this result, with the ascending chain condition.
- In this case " R " is also said to satisfy the ascending chain condition.
- An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.
- The same equivalence holds for infinite lattices which satisfy the ascending chain condition ( or the descending chain condition ).
- A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain condition.
- Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition.
- The lattice of subgroups of a group satisfies the ascending chain condition if and only if all subgroups of the group are finitely generated.
- A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps.
- Note that graded posets need not satisfy the ascending chain condition ( ACC ) : for instance, the natural numbers contain the infinite ascending chain 0.
- A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps.
- A finite lattice, or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric.
- By Zorn's lemma, this is equivalent to saying that the ascending chain condition holds : there is no infinite strictly ascending chain of congruences on " S ".
- The subgroup generated by the normal locally nilpotent subgroups is called the Hirsch Plotkin radical and is the generalization of the Fitting subgroup to groups without the ascending chain condition on normal subgroups.
- In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on " all " right ideals.
- More generally, any integral domain satisfying the ascending chain condition on principal ideals ( i . e . the "'ACCP "'), is an atomic domain.
- Many types of objects in abstract algebra can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called " spectrum of a Noetherian ring a Noetherian topological space.
- O . Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions.
- A GCD domain generalizes a unique factorization domain to the non-Noetherian setting in the following sense : an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals ( and in particular if it is Noetherian ).
- To give an example, let " A " be a ring that does not satisfy the ascending chain conditions on radical ideals, and put X = \ operatorname { Spec } A . " X " contains an open subset " U " that is not quasi-compact.
- That condition, called ascending chain condition on principal ideals or ACCP, is strictly weaker than the BFD condition, and strictly stronger than the atomic condition ( in other words, even if there exist infinite chains of proper divisors, it can still be that every " x " possesses a finite factorization ).
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