Question

13.12.8 Problem. Let R be a ring and, let M be an R-module. Let m be a nonnegative integer, and suppose that M1,..., Mm are R-submodules of M, and that M is the internal direct sum of M1,..., Mm. Let n be a nonnegative integer with n < m, and for each i E {1,...,n}, let N; be an R-submodule of M. Let N = N1 ++ Nn. ... (i) Prove that N is the internal direct sum of N1,..., Nn. (ii) Prove that M/N (M1/N1) .. (Mn/Nn) Mn+1 ® • • · ® Mm• (Hint: compose the map in $13.11.3 with a natural projection, and apply the First Isomorphism Theorem and $13.12.4.)

Answer 1

13 12.80) Givew M= M, M, Ⓡ O MW W=Wj + We t-. tNw where Nic Mi i=1,2..n are R-submodur. let ne N. Then we consider two representation of as ñ = ň, + Wat - tun ž., n? e Ni i = 1,2..n ñ ñ tñt.. tun we it and ñ . ,n e Mi l = 1,2..n . out as M is the internal derect sum of M2.. Man the representation of w is unique in M. Hence ŭ, až? *1 = 1,2.. n The representation of w is unique in n N= N + N₂ fr of Nw 3 N = N, N, .. (11) fet us define a map Ø : M (MIN.) 6 (M/W) / Nn Omni ... ooM mu . as follows Ø (met ma + muut tum ) = (n + Ny ) + (na +N2) + + (n +Nn) + mants to them Ther for any in the t heme nyt mat- & ut in and rer 0 (no, the + + room) fr (suy the t. & nun ) = (nu trim, t) +- + metru nt Win ) + (ment, trhu ant) + ff runt ruum) =/(m + y) + - + (unt Nu) + un nest- + nun ) + (no f Ng) + tment Nu) + munts to turn) = (n + un t. & num) + + Ø (mo, + nust- & uum) Hence 0 is a R-module homomorphism

More over for (in ts) + ( + N2) + ... + (mon + Nu) + maha, + + no E (M/W) 6 (M/W) 0 - 4 (Min) @ Muti 6. Mm. Ø ( ut, não .. nun même . .mm )= (myt Ny) + (17 + N2 )+ - + ( + Nn) + namesto 4mm Hence Ø is surjective . Hena by first is omorphism theorem M/Kery (M/N) (M/W) 6 (M/W) CG Mutio (DMM- wow kes ya? mix + et webine ma golu * - . tuetu) = INJ + N₂ + - +Nnto toto = ? m + un to them em metal, ) + 6M + Ne) ti & funt Nu) + montit + num = Wit Ng tot Watoto and = m+mot fimEM I miti ;. EN .=0 i=1,2.. n jantl... m = 2 m, then try to 40 EM ) MG EN; &i=1,2..ng = NON, © .. ON = N Hence M/N √ (M/W) & (M/W) + -- 0 MW /n) D MA 10 MM