General Matrix Inner Product?

This is inefficient and its is not precisely the set of arguments you requested but its 
concise and perhaps its close enough.

inner = function(a,b,f,g) apply(apply(outer(a,b,FUN=g),c(1,4),diag),2:3,f)
a=matrix(1:8,2,4)
b=t(a)
inner(a,b,sum,"*")

[,1] [,2]
[1,]   84  100
[2,]  100  120

a%*%b

[,1] [,2]
[1,]   84  100
[2,]  100  120

inner(a,b,all,"==")

[,1]  [,2]
[1,]  TRUE FALSE
[2,] FALSE  TRUE