Is the set of upper(and separately lower)-triangular matrices a ring?

Question

I was reading lecture notes which mentioned the set of upper (and separately lower)-triangular matrices of a certain dimensionality is a group under matrix multiplication. That made me wonder if they also form a ring under addition and multiplication.

So first, they are an abelian group under matrix addition:

• The sum of any number of triangular matrices is itself a triangular matrix.
• The 0 matrix is the 0 element.
• There is an additive inverse. (Element-wise negation)
• Matrix addition is commutative.

Then, they are a monoid under multiplication.

• The product of any number of triangular matrices is itself a triangular matrix.
• The identity matrix is the multiplicative identity.

And finally, multiplication distributes over addition.

Is that correct?

Answer 1

Yes. The reasoning above is correct.