In the Lam's book (A first course in noncommutative rings), he is representing the triangular ring with direct sum. I could not understand this part. How can we consider the triangular rings with direct sums , while we don't have the same multiplicative operation? I have tried to think about an isomorphism between triangular rings and direct sums as bimodules, but this won't help me to understand the ideals. Also, why do we need to consider the triangular ring by direct sum?