Question

How many non-isomorphic unital rings are there of order 4?

Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be a multiplicative identity? In the latter case assume the multiplicative identity to be (1,1). Some Theory from Lectures: A subring S of a general ring R is a subset that is also a ring with the same operations of addition and multiplication. To check that S is a subring it is enough to show that it is closed under subtraction and multiplication. An ideal S of a ring is a subring that is also closed under multiplication by a general element of the ring. In particular, a E S, b ER = ab, ba E S.

Answer 1

Let R be a ring of order 4. Men (R,+) is a commutative group of order 4. There are , upto isomorphism two possibilities for a group of order o. one is Zq and the other is Kriw Four group which is iromorphic to Z2 XZz Hello wot cau и с моst 2 now - isomorphic unital nigs Note that (zast.) is a oning with usual addittow and multiplicatou. For [as,[6] € 24 [a] + [b] = [ato [a].[6] = [ab] [2] [3] = [Q] = [s][a] [a] e Z4 Heuc It] is the id element. (6). Zq X Zz. Note that (Zeit,') is a sig. Hence (TqXZz, t, .) is also a rig where addition and multiplication n defined porùf woe. For ([]. [U] ) [cc] [d]) € Z2 X Zz ( [0],[] ) f(tc), [d]) = ([a+c], [utd] ). (LATIT). (c)2d1) =([ac]. Ibid)) ADO (ra), L v3). (1), L1T) - (Ta), Iv] )= (119,67).(19),LLI) Hena (1 !1,11]) is the unit id element, have exactly 2 now - isomorphic nuital sig upto isomor- Abo Heu ce we phism