Question

Let a and b be non-zero elements of a principal ideal domain R, and let 1 = (a) and I = (6). Show that the following are cquivalent: (i) I and I are comaximal. (ii) In J = II. (iii) ab is a least common multiple of a and b. (iv) 1 = ged(a,b).

Answer 1

a and b in.) ab is о common hiust of all, suppose hiren :- are non-zero elements of a principal ideal dornain R, let I = (a) and I = (b). To show :- The statements: it I and I are comaximal, is In T = II least multiple of a and b, in) 2a gedla b) que equivalent. that I and I are comaximal. Then I + J =R that is (a)+ (6) = R. As It I is the smallest ideal containing (a) and (b), it is the smallest ideal generated by the two non-zeuo elements, la,b). Then, we obsecure that U)= R=(a, b) = I + J = (a)+(b), so 1 is a greatest common divison of a and b. instead of supposing I and Tos comaximal, suppose that copume. Then, I is the greatest common dirison of of 'a' and 'b' which means that I is 'a generator for the smallest purcipal ideal containing 'a' and 'b. (a, b) = (1) = R. I are comaximal. Now, and b ane Then, we obseure It J = (a)+ (b) So I and

a go, it is clear that fore I and I to be comaximal, the greatest common dirison of 'a' and 'b' & must be t. ou vice-reusa. We know that 'a' and 'b' are non-zeuo elements of R. The least common multiple of'a' and 'b' is an element ecru'such that able and if a b le' then e le The ined (a) n (b) ou Inti non employ and its genematau, e is the least common multiple of and b. since, (a) (b) C (a)n(b), then ab Ela) (b) is non-remo. plo, we can say that canolb) in non-emply. TID, it has a gencuator e. from above, it can be said that e ela)n(b) implies that et la)(b) ou eE IT. e & lalnib) ou ecin with et IJ, we can say e e culb) implies that (e) cla) and eclb) which is tuwe only This shows that any two non-zero elements of a puinciple ideal domain have a least common multiple In the given question, the least common multiple of 'a' and '6' So, after doing the above steps, it is clear that all the given towe statements aule equivalent. On comparing that In JIJ if and ale and ble. is ab.