Answer to Question #141829 in Abstract Algebra for anjali

Question #141829
Let R be the ring and R^n= {(x1.......xn)/xi∈ R} be the R-module.
1] If I1,I2....In are ideal of R then N=I1xI2x....xIn={(x1.......xn)/xi∈ I} is a submodule in R^n
1
Expert's answer
2020-11-02T20:48:19-0500

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".


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