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

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)

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?

• Your reasoning looks correct to me. Jun 26, 2018 at 8:09
• All requirements for definition en.wikipedia.org/wiki/Ring_(mathematics)#Definition seem to be satisfied..See also en.wikipedia.org/wiki/Matrix_ring Jun 26, 2018 at 9:54
• You probably mean the set of upper triangular matrices, not the set of upper or lower triangular matrices.
– lhf
Jun 26, 2018 at 13:02
• If: Yes. That is what I was trying to get to with "unitriangular" matrices. The set of upper-triangular matrices of a particular dimensionality and separately, the set of lower-triangular matrices of a certain dimensionality. Jun 26, 2018 at 15:50