Ideals
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From: Arturo Magidin on 6 Aug 2010 14:17 On Aug 6, 3:57 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Thu, 5 Aug 2010, Arturo Magidin wrote: > > >>>> Thus, as commutativity hasn't been used, if R is a ring with unity and I > >>>> an ideal within R, then there's a maximal ideal J with I subset J. > > >>>> In addition, if I is a left ideal within R, then there's a maximal > >>>> left ideal J with I subset J and mutates mutandi for right ideals. > > >>> Offhand I'd guess that, if you want a maximal one-sided ideal, you can > >>> get by with just a one-sided multiplicative identity, instead of a 1. > >>> Um, right identities for left ideals and vice versa, if "left ideals" > >>> and "right identities" are defined the way I think they are. > > >> If there is no identity, then there are either multiple left or > >> multiple right identities. > > > False. > > It's correct in the context of rings with identities. Right: if you add hypothesis, then you may change a false statement into a correct one. Nonetheless, your statement was false. It is *definitely* not true that a ring with no identity must have multiple left or multiple right identities. > > > Pick your favorite Abelian group G with more than two elements, > > written additively, and define a multiplication * on G by a*b = 0 for > > all a and b in G. > > > Kindly exhibit the "multiple left" or "multiple right" identities of > > the ring (G,+,*). > > { } Duh. -- Arturo Magidin
From: Arturo Magidin on 6 Aug 2010 14:19 On Aug 6, 4:00 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Thu, 5 Aug 2010, Butch Malahide wrote: > > On Aug 5, 4:40 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > > >> I is a left ideal within a ring R when -I, I + I, IR subset I. > > > No, that's a *right* ideal. I quote from p. 49 (beginning of section > > 16 if you have a different edition) of B. L. van der Waerden, Modern > > Algebra, Volume 1, revised English edition (in the book o and m are > > small German letters): > > Buzzards. What about left and right identities? How are they defined? The obvious way. If R is a ring, and a in R, then a is a left identity if and only if for every x in R, ax = x. Your problem was that when you look at IR, you are multiplying every element of I by an element of R *on the right*; if your I is a left ideal, then you have no warrant for thinking that ax will lie in I when a is in I and x is in R. -- Arturo Magidin
From: Arturo Magidin on 6 Aug 2010 14:27 On Aug 6, 3:55 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > >> Thus my unchallenged claim that Rf has no maximal > >> ideal is dismissed as hallucinated illusion, for > >> the usual Zorn's lemma proof holds. > > > No; the Zorn's Lemma proof does not work for Rf, because you do not > > know whether the union of C will be a *PROPER* ideal. > > For example. > Define r_n:N -> Z_2 with r_n(k) = 1 if 1 <= k <= n, = 0 otherwise.. > C = { (r_n) | n in N\1 } is a chain of ideals of Rf. A chain of *proper* ideals. Yes. > \/C = Rf > > >> Now however, if instead of Rf, the ring is Ra = (Z_2)^N, > >> then still, doesn't > >> \/C = { r in (Z_2)^N | r(1) = 0 } /\ Rf? > > > Yes, but this is no longer a maximal ideal of Ra. So what? > > It was a set theory exercise. And we were supposed to guess? You do remember you aren't just thinking these words, you are actually broadcasting them, right? > >> In addition, if I is a left ideal within R, then there's a maximal > >> left ideal J with I subset J and mutates mutandi for right ideals. > > No... Really? > > Yes. Recall that this is in addition to conclusion above from the > premise that R has an identity. Can't tell sarcasm when it's staring you in the face either, huh? > Are definitions regarding left and right ideals and > left and right identities the same throughout all texts? Yes. > Are these correct or crossed? > > A subring I, of a ring R is a left ideal when RI subset I > a right ideal when IR subset I. Modulo your traditionally clipped syntax, yes, they are correct. > An idempotent d of a ring R is > a left identity when for all x, dx = x > and > a right identity when for all x, xd = x, Correct, though you do not need to assume idempotency. -- ArturO Magidin
From: William Elliot on 7 Aug 2010 03:37 On Fri, 6 Aug 2010, Arturo Magidin wrote: > On Aug 6, 4:00�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >> On Thu, 5 Aug 2010, Butch Malahide wrote: >>> On Aug 5, 4:40�am, William Elliot <ma...(a)rdrop.remove.com> wrote: >> >>>> I is a left ideal within a ring R when -I, I + I, IR subset I. >> >>> No, that's a *right* ideal. I quote from p. 49 (beginning of section >>> 16 if you have a different edition) of B. L. van der Waerden, Modern >>> Algebra, Volume 1, revised English edition (in the book o and m are >>> small German letters): >> >> Buzzards. �What about left and right identities? �How are they defined? > > The obvious way. If R is a ring, and a in R, then a is a left identity > if and only if for every x in R, ax = x. > > Your problem was that when you look at IR, you are multiplying every > element of I by an element of R *on the right*; if your I is a left > ideal, then you have no warrant for thinking that ax will lie in I > when a is in I and x is in R. If I is a right ideal of R, ie IR subset I and d is a left identity, ie for all x, dx = x, then the right ideal (d,I) = R and there's a maximal right ideal J with I subset J.
From: William Elliot on 7 Aug 2010 03:58 > Intesrestingly, while your assertion was false, it is true that a ring > with more than one left identity has infinitely many of them. This is > a problem in Jacoboson's 3-part tome on algebra, and I did this a very > long time ago. It was a fun problem, and I recommend it to everyone > as an interesting problem to work on. If there's a unique left identity, then it's the identity. If there's a left and a right identity, then they're both the identity. The identity is uniquely both a right and a left identity. If there's two left identities, and the ring has no torsion elements, then there's infinitely countable many left identities. What did you do for rings with torsion elements? In addition, your problem implies that no finite ring can have two left identities. Is that true? BTW, why are torsion elements called torsion elements? Because some how they twist the ring?
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