Question

This is abstract algebra, about rings.

29. Let A be any commutative ring with identity 1 + 0. Let R be the set of all group homo- morphisms of the additive group A to itself with addition defined as pointwise addition of functions and multiplication defined as function composition. Prove that these operations make R into a ring with identity. Prove that the units of R are the group automorphisms of A (cf. Exercise 20, Section 1.6).

Answer 1

29) let A be any commutative sing with unity 170 Cidentitt let R be the set of all group homomorphisms of the additive group A to itself with addition defined as pointmise addition of funnst multiplication defined of functicos composition. i.e. R=2 fiatal f is group homomorphism? & for any f, fz e R addition of fitte is (fi+fe) can = ficant ofxca) 4 multiplication multiplication office is (fi• $] (a) = Cleopa) (@=fi (fo.ca)) Claim : R is ring with identity. R: 1st we show R is Abelian group : let fi ifa ER flet at A pre fi, fa are group homomophispy from A to A consider, (fit f) (a) & fical+ Faca) --given addition Consider Gfit fe] (aitaz) = fi cautaz) + fa cartaz) = fican+f, C42l+f2ca1) 472592) =Litfs) Caitffit ife) (42) :. fit h is homomorphism from A to A. > fit to ER : ZO

por fifa f ER (fit fra) +33 = fit (fatfs) holds clearly. -☺ row, for amt FER 3 af Erst zero map Ō ER Sit. Cf+ o) ca)= faltāca) & õ is homomorphism from A to A. f õisidential ő a ker & is Ô ertimal now for ant ft R, 9 -f ER sit a (f+(F)] cano fra) +(-+) (a) = f(a) - Fla) = 0 L ti:Cf- f)(a) sos kerø = grow, fit fe)(a) = fzt filca) group urt to C by 0,0,0, 00) R2: we shove multiplication is associative let f. for f3 ER & at A (fi fali Ps) ca) = ((frota2083) ca) =( (F. (fr)) Is) (a) = fi (felfaca))). Eti C42063) Cai) = 10 (F2013)(a) alfi. Cf2. fs))ca) Hence, R is commutative

2 . multiplication in R is associative. Roi noue we show distributive laws holds. let fi, fs, f3 ER & at A. ((fit fra). fo) cas = ((fit fe)• ) (a) = , ofs)+($20f))(a) = ((fref+ (82°Fs)) (a) = ('#s)ca) + (fritz)ca) [fi+fz). 63- frets + Facts Similiary, fio (fz+73) = fr Fz + fif3 ..pon from R., R2 R3 R is ring. Now, for identity map I i A A , I is homomorphism from A to A defined by I (a) =a. :. For f GR, (f. I) (a) = (fo I) (a) = 767 (a)) = f(a) ; . I is multiplicative identity of R. Hence R is ring with identity I.

Automorphism means homomorphism from group A to itsef which is H 4 onto. It is given that for fER, f is homomorphism from A to A. coiffie to be is t &conto. Hence, f exists ...(f is 1-1) & Fiano is group homomorphism. - fER .. units of R are group automorphismy of A.