Modulo Of Rota-Baxter Algebras With Nonzero Weights

In the 1960s,Rota-Baxter algebra originated from random theory,which consists of an associative algebra and a linear operator.It has been widely concerned by many mathematicians and physicists,and has been used in many fields such as algebra and physics.In this paper,we study the modules of polynomial RotaBaxter algebras(k[x],P)of weight nonzero and the free commutative non-unital Rota-Baxter algebra(k*[x],P)which is the algebra of polynomials in one variable without constant term with Rota-Baxter operators of nonzero weight.The main result shows that every module over a non-zero-weighted Rota-Baxter algebra(k[x],P)and(k*[x],P)is equivalent to the modules over a noncommutative algebra generated by two indeterminates with a generation relationship.Furthermore,we provide the classification of modules of non-zero-weighted Rota-Baxter algebras(k[x],P)and(k*[x],P)through solution to some matrix equations.In addition,we study the irreducible and indecomposable modules of(k*[x],P).Finally,we give some examples of modules of(k*[x],P).