Introduction to Abstract Algebra, Solutions Manual, 4th Edition

7.1 Modules

1.

a. 0x = (0 + 0)x = 0x + 0x, so 0x = 0.

c. Using (a), x + (− 1)x = (1 + (− 1))x = 0x = 0 = x + (− x) . Now “subtract” x from both sides by adding −x to both sides.

2.

a. If α : M → N is onto and R-linear, and if M = Rx₁ + + Rx_n, then N = Rα(x₁) + + Rα(x_n) . Since some of the α(x_i) may be zero, the result follows.

c. Let K = Rx₁ + + Rx_m and let M/K = R(y₁ + K) + + R(y_n + K) where the x_i and y_j are in M. If x M let

with r_i R for each i. Then x − (r₁y₁ + + r_ny_n) is in K, so

where s_j R for each j. Hence {x₁, . . ., x_m, y₁, . . ., y_n} generates M.

3. If Re is an ideal ...