If-0-M-f-M-g-M-0-is-a-short-exact-sequence-and-M-M-are-two-finitely-generated-R-modules-then-prove-M-is-finitely-generated-Hint-f-g-are-two-R-ho

Question Number 190784 by mnjuly1970 last updated on 11/Apr/23

If, 0 ⇢ M^{'} \overset{f}{⇢} M \overset{g}{⇢} M^{″} ⇢ 0 is

a short exact sequence and M^{'}, M^{″} are

two finitely generated R - modules

then prove M is finitely generated .

Hint : f, g a r e t w o R - h o m o m o r p h i s m .