#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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Changes (by tscrim):
* status: needs_info => needs_work
Old description:
> Should Zero morphism be a morphism of rings?
> {{{
> sage: H= Hom(QQ, QQ)
> sage: f=H(0)
> Traceback (most recent call last):
> File "<stdin>", line 1, in <module>
> File "_sage_input_4.py", line 10, in <module>
> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8
> -*-\\n" +
> _support_.preparse_worksheet_cell(base64.b64decode("Zj1IKDAp"),globals())+"\\n");
> execfile(os.path.abspath("___code___.py"))
> File "", line 1, in <module>
>
> File "/tmp/tmplncYUJ/___code___.py", line 3, in <module>
> exec compile(u'f=H(_sage_const_0 )
> File "", line 1, in <module>
>
> File "/home/kamalakshya/sage/local/lib/python2.7/site-
> packages/sage/rings/homset.py", line 184, in __call__
> raise TypeError("images do not define a valid homomorphism")
> TypeError: images do not define a valid homomorphism
> }}}
>
> In other words, do we assume that our homomorphisms take 1 to 1?
New description:
Currently the zero morphism cannot be constructed starting from `QQ`:
{{{
sage: H= Hom(QQ, QQ)
sage: f=H(0)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_4.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8
-*-\\n" +
_support_.preparse_worksheet_cell(base64.b64decode("Zj1IKDAp"),globals())+"\\n");
execfile(os.path.abspath("___code___.py"))
File "", line 1, in <module>
File "/tmp/tmplncYUJ/___code___.py", line 3, in <module>
exec compile(u'f=H(_sage_const_0 )
File "", line 1, in <module>
File "/home/kamalakshya/sage/local/lib/python2.7/site-
packages/sage/rings/homset.py", line 184, in __call__
raise TypeError("images do not define a valid homomorphism")
TypeError: images do not define a valid homomorphism
}}}
--
Comment:
For future reference, it's better to just put these types of questions as
comments.
Replying to mkamalakshya:
> Should Zero morphism be a morphism of rings?
> In other words, do we assume that our homomorphisms take 1 to 1?
We have to be careful about what we mean by '1' in the image. In
particular, the image is the trivial ring (field) with `0 = 1` and it
still satisfies all of the usual ring (field) properties (where we aren't
dividing, but it's just a statement about the multiplicative inverse). So
in this case, the image of 1 from `QQ` is `0` and satisfies all of the
expected multiplicative identity axioms:
* `0x = x0 = x` (note that `x = 0` in the image)
* `x x^{-1} = x^{-1} x = 0` (and `x^{-1} = 0` as well in the image)
So the zero morphism is a morphism as rings (fields) by sending the
additive/multiplicative identity to the additive/multiplicative identity.
--
Ticket URL: <http://trac.sagemath.org/ticket/16843#comment:6>
Sage <http://www.sagemath.org>
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and MATLAB
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