Let "R" be the ring and "R^n= \\{(x_1.......x_n)|x_i\\in R\\}" be the "R"-module.
Let us show that if "I_1,I_2,...,I_n" are ideals of "R" then"N=I_1\\times I_2\\times\\cdots \\times I_n=\\{(x_1,...,x_n)|x_i\\in I\\}" is a submodule in "R^n".
Taking into account that "I_1,I_2,...,I_n" are ideals of a ring "R", we conclude that "I_1,I_2,...,I_n" are additive subgroups of a ring "R", and therefore, "N=I_1\\times I_2\\times\\cdots \\times I_n" is a additve subgroup of "R^n=R\\times R\\times\\cdots \\times R". Let "r\\in R" and "(a_1,...,a_n)\\in I_1\\times I_2\\times\\cdots \\times I_n" be arbitrary. Since "I_1,I_2,...,I_n" are ideals of a ring "R", "ra_1\\in I_1,..., ra_n\\in I_n" . Thus, "r(a_1,...a_n)=(ra_1,...,ra_n)\\in I_1\\times I_2\\times\\cdots \\times I_n" , and we conclude that "N" is a submodule in "R^n".
Comments
Leave a comment