  Several operations can be defined on the set of all linear recurrent
sequences, such as the binomial convolution (Hurwitz product) or the
multinomial convolution (Newton product). Using elementary techniques, we prove
that this set equipped with the termwise sum and the aforementioned products
are R-algebras, given any commutative ring $R$ with identity. Moreover, we
provide explicitly a characteristic polynomial of the Hurwitz product and
Newton product of any two linear recurrent sequences. Finally, we also
investigate whether these $R-$algebras are isomorphic, considering also the
R-algebras obtained using the Hadamard product and the convolution product.
