I have found that in most books on homological algebra, the author(s) work in the general setting of noncommutative rings. This leads to several irritating complications. For example, the tensor product $M\otimes_R N$ only makes sense if $M$ is a right $R$-module and $N$ is a left $R$-module.
Personally I am completely uninterested in noncommutative algebra. Because of this, whenever I have studied homological algebra in the past I simply ignored these details and mentally assumed all rings are commutative.
However, it has occurred to me that perhaps I may be crippling myself by doing this. There are certainly many places in mathematics where considering a more general setting provides richer information about the specific setting one is interested in. For example, allowing nilpotent coordinate rings in algebraic geometry allows one to construct useful maps to or from the integral domains that one is interested in, which can give otherwise unavailable information.
Generally speaking, I want to ask for some examples where being aware of the intricacies of noncommutative homological algebra is important for one interested in the study of commutative rings.