Let $R_p := \{R \in K^{n \times n}: r_{ij} = 0,\ \mathrm{if}\ i > j - p\}$, how to show that the set of right triangle matrices $R_0$ paired with matrix addition and matrix multiplication forms a ring with 1/unital ring?
Asked
Active
Viewed 68 times
0
-
2Welcome to Math.SE! What have you tried so far? What is the definition of "zero ring" which you are using? – hasManyStupidQuestions Dec 27 '19 at 22:55
-
@hasManyStupidQuestions per the first paragraph of this wiki article, I would guess that a zero ring in this context is a rng of square zero, i.e., a rng in which $xy = 0$ for all $x$ and $y$. – Ben Grossmann Dec 27 '19 at 23:07
-
Actually this definition doesn't make the statement to be proved true, so presumably the definition is a bit different – Ben Grossmann Dec 27 '19 at 23:11
-
I actually meant a unital ring.. Edited the question. – Dec 27 '19 at 23:22
1 Answers
0
We can show that $R_0$ is a unital ring by applying the subring test. Clearly, $R_0$ is closed under addition and subtraction. By your earlier post, $R_0$ is closed under multiplication.
In fact, we can say a bit more: the inverse of an upper-triangular matrix is upper-triangular. So, a matrix $M \in R_0$ is a unit of $R_0$ if and only if it is a unit of $K^{n \times n}$.
Ben Grossmann
- 234,171
- 12
- 184
- 355