The set of linear recurrent sequences can be endowed with several operations such as the binomial convolution or the multinomial convolution (also known respectively as the Hurwitz product and as the Newton product). We prove, applying elementary techniques, that this set equipped with the termwise sum and the aforementioned products are $R-$algebras, for 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. We also explore whether these $R-$algebras are isomorphic, considering also the $R-$algebras obtained using the Hadamard product and the convolution product. Finally, we deal with some well-known and important operators that act on sequences and their relationships with the aforementioned products.

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