Question

Let R be a ring, let S be a subring of R and let' be an ideal of R. Note that I have proved that (5+1)/1 = {5 +1 | 5 € S) and I defined $:(5+1) ► S(SO ) by the formula: 0/5 + 1)=5+(SNI). In the previous video I showed that was well-defined. Now show that is a ring homomorphism. In other words, show that preserves both ring addition and ring multiplication. Then turn your work into this dropbox.

Answer 1

Let us s/sni TO show Take, tSt I Then Let & be aring, s be a subring and I be an ideal of R. de fine 02 (SI) - by $(8+ I)- 5+ (S0]) where ses. & is a sing homomorphism. Sif I and S, & I where, S., ES $ ( 61+1) + €2+ I)) ${{1+52) + I) (: by defination (a +1 ) + (673)=(a+b) +1) ($+32) + (501) vaibe R (S, +501) +($z +593) *(S,+1) + $(5+5). $( {$i+ I ) + (s, + )) 6(S145) +$1544) Kit I), (SI) Now, ( (8+2). (8, + )) = f(ex S3) + 1) ((a+ I). (bf I)= (as) + I, & a, bER) (S, S2) + (SI) (51 +610.5)) (S2+ (8nI)) - Ørs, +1) 8182+1). Pls, 41).(S, +33) = $(5+3). 675 +5) Hence & is asing homomor phism. 2 ESTI. 2

-لها Toska, (s, + I) = (S, + I) mo there si, S₂ES. (su+1) = *(5,+5) (for well de fried mers) To Show Then Sit I - S, & I (si - S₂) & I. and also, (1-S2) ES ... (S - SP) 4 SOT . = S, +(SNI) = S₂ +8 NI) -> $( 51 +1) = $($2+1). e is well defined Hence arab xrow
I hope you know that Φ is well defined.In case you don't know I have shown the well definedness of Φ. So if wish to write it freshly you first show the well defined ness then show that it is a ring homomorphism