Timeline for vectors with entries from a finite ring
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 24, 2010 at 16:13 | answer | added | Greg Kuperberg | timeline score: 1 | |
| Oct 10, 2010 at 21:56 | comment | added | felix | Kevin, this is indeed true. $v_1, \dots, v_n \in \mathcal{R}^n$ give a basis of $\mathcal{R}^n$ iff $\det (v_1, \dots, v_n) \in \mathcal{R}^\ast$, and this is the case iff $\det (v_1, \dots, v_n) \neq 0$ modulo every maximal ideal of $\mathcal{R}$. Hence, the probability that a random $n \times n$-matrix is invertible is the product of the probabilities for random $n \times n$-matrices to be invertible over all residue fields of $\mathcal{R}$. | |
| Oct 10, 2010 at 8:55 | comment | added | Kevin Buzzard | I've not thought too hard about this, but: a general finite ring is semi-local so a product of local rings, so that reduces the question to local rings. And then via a Nakayama argument my gut feeling is that you'll be able to reduce to the case of the residue field, which you've already done. | |
| Oct 10, 2010 at 6:29 | answer | added | Aaron Meyerowitz | timeline score: 0 | |
| Oct 10, 2010 at 4:42 | history | asked | Sarah | CC BY-SA 2.5 |