#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 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 don't like that enlarging the codomain changes whether the function is
a morphism or not.
Well, that is what you get from the definitions! This is related to the
fact that "enlarging the codomain" also changes which element is the
multiplicative identity. However, it is important to note that in this
case "enlarging the codomain" is misleading terminology in the ring-
theoretic context, as opposed to the set-theoretic (or module-theoretic)
context, for the following reason.
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.)
--
Ticket URL: <http://trac.sagemath.org/ticket/16843#comment:11>
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