Question

QUESTION 4 (a) Let RS be a ring homomorphism with I an ideal of R and J an ideal of S. Define 0(I) = {$(1) I ET) and o-'(J) = {ve R(y) € J} and check as to whether or not (i) °(1) is an ideal of S (6) (ii) o-'() is an ideal of R (6) (Hint: I, J are two-sided ideals and in both cases of (i) and (ii) above, first check the subring conditions) (b) Given a ring R' with K an ideal(two-sided) and H a subring, then check as to whether or not Kn is an ideal(two-sided) of H (6) Fint: first check the subring conditions) (c) Define a division ring and using that definition (of a division ring), define a field

Answer 1

then Sola: (a) i) Let I be ideal R of & (I) is not ideal of S Because Take R = 7 is S=Q ring homo morphism of Z Now, Take 1= 27 then clearly I= 24 is ideal but Q (I) = 0 (27) = 27 is not ideal of Because, % EP s 4 26 27 then if oz is ideal of a 16L k which is contradiction 22 As 1624-001) o 1) is not ideala an of := Claim : Is I is ideal is an ideal of S then $'(0) R of Let 4,99 € *"'() { re R then Top.to Yo + yo '(I) { ry, € $'() As Yi, Yg € 0 "(I) » F xi, R2 E J such that $ (4) = 2 (42) = 22 As (4) + f2) = 2, +22 € J 6 (9, +42) Yothaf 8" (7) of S (6 As I is ideal

Also, o cryo) $(r) 0 14,) = 0 (0) 2, EJ ry, E P (J) Thus 9" (J) is an ideal of R. b tet Ŕ be a ring H is a subring ring with k be an ideal f Claim KOH is an ideal of let y, , У, Є КОН + rєН t.pt. Yo+Y2Eknu € 89, EKOH As 9,92 EKOH Y, Y, Ek & Yo, I₂ EH Yo ty, Ek and Yety, EH • As kis ideal & His a subring) yet 42 € KOH Also, "IEK & rEHOR ry. Ek (iki edeal and y, EH & reh ry, EH ( His subring of R! Thus ry, EKOH ¥ reH f g EkoH Hence, KOH is subring of H. of R © Division Ring A ring R 0 is called division ding if every mongero element has multiplicatire inverse. Field: A commutative division ring is called as Field