Triangular rings with direct sums

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?

• I don't understand what you mean by representing a triangular ring with direct sum. – Qiaochu Yuan Apr 21 '13 at 23:20
• Triangular rings : 2x2 matrix , its entries are the rings R( ring ), M (R,S) bimodule, S(ring) and the direct sum of R + M + S – Bbbh Apr 21 '13 at 23:31
• The direct sum is as abelian groups. Then you define a multiplication on the corresponding abelian group. – Qiaochu Yuan Apr 21 '13 at 23:33
• How, you mean individually. Could you clarify it? – Bbbh Apr 21 '13 at 23:35
• I'm not sure what you're confused about. – Qiaochu Yuan Apr 21 '13 at 23:36