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.
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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 ".
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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.
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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.
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More generally, any integral domain satisfying the ascending chain condition on principal ideals ( i . e . the "'ACCP "'), is an atomic domain.
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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.
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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.
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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 ).
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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.
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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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