#16843: Zeromorphism
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Reporter: mkamalakshya | Owner: mkamalakshya
Type: defect | Status: needs_work
Priority: minor | Milestone: sage-6.4
Component: algebra | Resolution:
Keywords: days60 | Merged in:
Authors: Kamalakshya Mahatab | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Comment (by tscrim):
Replying to [comment:11 pbruin]:
> Hi Travis,
> > I agree that there would be problems if the 1,,R,, did not go to
1,,Im(f),, under a funtion `f` and that we cannot show this from the ring
axioms (unlike fields).
> As far as I can see, there is no difference between fields and other
rings here.
I was just saying for fields, this is not something you need to assume, it
can be shown from the axioms.
> The intuiton behind your objection seems to rely on an implicit
assumption that the zero ring is a subring of any other ring. In fact
this is not the case. Namely, completely in parallel to the definition of
ring homomorphisms, a subring of a ring ''R'' is an additive subgroup
closed under multiplication ''and containing the multiplicative identity
element of R''. Equivalently, subrings are precisely the images of ring
homomorphisms. This implies that the zero ring is not a subring of any
ring except itself. (On the other hand, the zero ring admits a trivial
homomorphism ''from'' any ring.)
Ah I see, I had the wrong definition in my mind. I agree with you now, we
should instead fix the other constructions.
--
Ticket URL: <http://trac.sagemath.org/ticket/16843#comment:12>
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