I don't think that makes sense. Ideals have no natural multiplicative unit,
whereas a type of ring I tend to think of has (admittedly, this may
sometimes be relaxed).
I'm not sure I see your point here; indeed, an ideal of ring with a
multiplicative unit need not necessarily include the unit, but this
doesn't make it less of a ring.
In my book it does. Of course there are non-commutative rings, and rings
without units, but for me "ring" without further qualifications means
"unital commutative ring, with homomorphism respecting the identity".
And while I agree that considering rings without a multiplicative
identity probably makes sense sometims, I don't think any commutative
algebraist will be particularly happy with a framework where ideals are
non-unital subrings...
[I agree with the rest of your mail.]
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