Question

Let (I,+, ·) be an integral domain. Consider the set {(a,b) : a,b ? I and b ? 0_I}. There is an equivalence relation on this set given by (a,b) ? (c,d) if ad = bc. Let [(a,b)] denote the equivalency class of (a,b). Define R := {[(a,b)] : a,b ? I and b ? 0_I} and • [(a,b)] + [(c,d)] = [(ad + bc, bd)]; • [(a,b)] · [(c,d)] = [(ac, bd)]. It is known that (R, +, ·) is a ring. I want you to check some of the ring axioms below in addition to other properties that R might have. Hint: Think about this in terms of I = ? and R = ? for an example. (a) Show that addition + is associative. (b) What is 0_R? (c) What is 1_R? (d) Show that multiplication · is commutative. (e) Show that the nonzero elements of R are units (f) Conclude that every integral domain is a subring of some field.

          Let (I,+, ·) be an integral domain. Consider the set {(a,b) : a,b ? I and b ? 0_I}. There is an equivalence relation on this set given by (a,b) ? (c,d) if ad = bc. Let [(a,b)] denote the equivalency class of (a,b). Define R := {[(a,b)] : a,b ? I and b ? 0_I} and

• [(a,b)] + [(c,d)] = [(ad + bc, bd)];
• [(a,b)] · [(c,d)] = [(ac, bd)].

It is known that (R, +, ·) is a ring. I want you to check some of the ring axioms below in addition to other properties that R might have. Hint: Think about this in terms of I = ? and R = ? for an example.

(a) Show that addition + is associative.
(b) What is 0_R?
(c) What is 1_R?
(d) Show that multiplication · is commutative.
(e) Show that the nonzero elements of R are units
(f) Conclude that every integral domain is a subring of some field.

Let (I,+, ·) be an integral domain. Consider the set (a,b) : a,b ? I and b ? 0I. There is an equivalence relation on this set given by (a,b) ? (c,d) if ad = bc. Let [(a,b)] denote the equivalency class of (a,b). Define R := [(a,b)] : a,b ? I and b ? 0I and

• [(a,b)] + [(c,d)] = [(ad + bc, bd)];
• [(a,b)] · [(c,d)] = [(ac, bd)].

It is known that (R, +, ·) is a ring. I want you to check some of the ring axioms below in addition to other properties that R might have. Hint: Think about this in terms of I = ? and R = ? for an example.

(a) Show that addition + is associative.
(b) What is 0R?
(c) What is 1R?
(d) Show that multiplication · is commutative.
(e) Show that the nonzero elements of R are units
(f) Conclude that every integral domain is a subring of some field.

Added by Shawn B.

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