Timeline for Is every locally free module of rank $1$ over a commutative ring concretely invertible?
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| Jan 2, 2020 at 1:05 | comment | added | Jesse Elliott | Note also that a ring is said to be Marot if every ideal containing a non-zerodivisor is generated by non-zerodivisors. Marot rings are therefore rings with "lots" of non-zerodivisors. It is known that every ring with few zerodivsors is Marot. Intuitively, this means that a ring with few zerodivsors has lots of non-zerodivsors. This gives some further justification for the terminology. | |
| Jan 1, 2020 at 14:20 | comment | added | Jesse Elliott | I think the condition that the total quotient ring be semilocal is more intuitive than the set of zerodivisors being a finite union of primes, so that's why I gave the former definition instead of the latter (which is the standard definition given in references). | |
| Jan 1, 2020 at 14:12 | comment | added | Jesse Elliott | Def: A ring $A$ has few zerodivisors if the the set of all zerodivisors of $A$ is a finite union of primes. From prime avoidance and the fact that the set of all zerodivisors of any ring is the union of the primes that are maximal with respect to not containing a zerodivisor, it follows that a ring has few zerodivisors if and only if its total quotient ring is semilocal. Being able to represent the set of zerodivisors as a finite union of primes is a finiteness condition on the set of zerodivisors, hence the terminology. | |
| Jan 1, 2020 at 13:46 | comment | added | Gro-Tsen | You speak of rings with “few zero divisors” as being equivalent to $\operatorname{Quot}(A)$ being semilocal: is this a definition of having “few zero divisors”? If so, what is the logic behind this terminology? And if not, what is the definition? | |
| Dec 30, 2019 at 14:22 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Dec 27, 2019 at 13:38 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Dec 27, 2019 at 13:31 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Dec 23, 2019 at 0:10 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Dec 22, 2019 at 3:34 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Dec 22, 2019 at 3:21 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Dec 22, 2019 at 1:42 | history | edited | Jesse Elliott | CC BY-SA 4.0 |
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| Dec 22, 2019 at 1:07 | history | answered | Jesse Elliott | CC BY-SA 4.0 |