If f is a matrix over R, and v is a 1 by k matrix over another ring S, then the result is obtained by substituting the entries in v for the variables in R.
If f is a module over R, then substitution amounts to substitution in the matrices of generators and relations defining the module. This is not the same as tensor product!
If v is a ring, then the result is obtained by substituting the variables of v for the variables of R with the same name. The substitution extends to the coefficient ring of R, and so on.
If v is a list of options {a => f, b => g, ...} then the variable a is replaced by the polynomial f , etc.
i1 = R = ZZ/101[x,y,z]
o1 = R
o1 : PolynomialRing
i2 = f = x+2*y+3*z
o2 = x + 2 y + 3 z
o2 : R
i3 = substitute(f,{x=>x^3, y=>y^3})
3 3
o3 = x + 2 y + 3 z
o3 : R
i4 = S = ZZ/101[z,y,x]
o4 = S
o4 : PolynomialRing
i5 = substitute(f,S)
o5 = 3 z + 2 y + x
o5 : S
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