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It is well-known that the Steenrod algebra $A$ is self-injective as a gradedring. We make the observation that simply changing the grading on $A$ can makeit cease to be self-injective. We see also that $A$ is not self-injective as anungraded ring.  These observations follow from the failure of certain coproducts of injective$A$-modules to be injective. Hence it is natural to ask: which coproducts ofgraded-injective modules, over a general graded ring, remain graded-injective?We give a complete solution to that question by proving a graded generalizationof Carl Faith's characterization of $\Sigma$-injective modules. Specializingagain to the Steenrod algebra, we use our graded generalization of Faith'stheorem to prove that the covariant embedding of graded $A_*$-comodules intograded $A$-modules preserves injectivity of bounded-above objects, but does notpreserve injectivity in general.

The Steenrod algebra is self-injective, and the Steenrod algebra is not self-injective