Chain conditions satisfied by some algebraic structures including ideals in certain commutative rings From http://en.wikipedia.org/wiki/Ascending_chain_condition:
"In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings.[1][2][3]
Notes
1 ^ Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.
2 ^ Fraleigh & Katz (1967), p. 366, Lemma 7.1
3 ^ Jacobson (2009), p. 142 and 147

References
    Atiyah, M. F., and I. G. MacDonald, Introduction to Commutative
    Algebra, Perseus Books, 1969, ISBN 0-201-00361-9
        Michiel Hazewinkel, Nadiya Gubareni, V. V. Kirichenko. Algebras, rings and modules. Kluwer Academic Publishers, 2004. 
    ISBN 1-4020-2690-0

        John B. Fraleigh, Victor J. Katz. A first course in abstract algebra. Addison-Wesley Publishing Company. 5 ed., 1967. 
    ISBN 0-201-53467-3

        Nathan Jacobson. Basic Algebra I. Dover, 2009. ISBN 978-0-486-47189-1"

I have the 7th edition of Fraleigh and Katz but can't find the equivalent reference.  I'm nearly sure that in another Wikipedia page it spells out what kind of ideals and commutative (semi?)rings but I can't find it any more (tab deleted somehow and can't get it back).  It also spells out why this is true.  Can anyone please point me to any references?
Thanks very much.  (Sorry about the references to Wikipedia pages.)
 A: The ACC and DCC are conditions on a poset, not on the elements of the poset. For most algebraic objects, the important subobjects form a poset under inclusion. There are examples of rings, modules, groups and semirings which do have and which do not have the ACC and/or DCC.
Commutative rings whose ideals satisfy the ACC are called Noetherian rings. Commutative rings whose ideals satisfy the DCC are called Artinian rings.
I'm pretty sure the identical definitions are carried over for ideals of semirings.
For noncommutative rings, there is a notion of right/left Artinian/Noetherian based on the lattice of right ideals and the lattice of left ideals.
Groups with the ACC on subgroups are called Noetherian, and groups with the DCC on subgroups are called Artinian. The same goes for modules and their submodules.
More generally lattices, which are also posets, can be labeled Artinian/Noetherian according to whether or not they have the DCC/ACC on their members.
Bonus: There is one freaky thing you might like, however. It turns out that for rings, if the ring satisfies the DCC on right ideals, it also satisfies the ACC on right ideals. This is the Hopkins-Levitzki theorem, if you ever have the time to study it :) It's false for groups and modules. I'm not sure if it's false for semirings or not!
Non algebraic example: A topology is called Noetherian if it satisfies the ACC on open sets. I've never actually seen Artinian topological spaces studied, but a quick google search yields hits, so maybe that exists too.
A: For example, you can check these notes.
