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Show that a set of matrices is a ring without an identity element.

I am given the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. Also, the matrix has entries a_ij, where i and j are natural numbers and there is a natural number n such that a_ij = 0 if i >= n or j >= n.

I need to understand how to show that this set of matrices is a ring without identity element.

Solution Preview

Proof:
Suppose the set of infinite-by-infinite matrices with real entries that have only finite many nonzero entries is R.
First, I show that R is a ring.
For any two elements A=(a_ij) and B=(b_ij), we have the following observations.
1. A+B=(a_ij+b_ij)=(b_ij+a_ij)=B+A is also an infinite-by-infinite matrix. Since A and B have finite many nonzero entries, assume A has m nonzero entries and B has n nonzero entries, then A+B has at ...

Solution Summary

It is shown that a set of matrices is a ring without an identity element. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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