Answer to Question #294452 in Abstract Algebra for Efiii jaan

Question

Show that a ring can't be expressed as union of 2 proper ideals but it is possible to express it as a union of three proper ideals

Answer 1

Proper ideal of a ring is a subgroup of this ring which is not equal to the ring itself. If the group (ring) R can be expressed asa union of 2 subgroups, A and B, imagine that there are two elements, x belongs to the difference A\B, and y - to the difference B\A. Hence the product z=x*y can belong to A or B - consider 2 cases. In the first case both z and x belong to A because x belongs to A\B and z=x*y. So we have y=z/x belongs to A, but y can not belong to A because y is the difference B\A. And we have the symmetrical situation for the other case - z=x*y belongs to B. So both z and y belong to B because y belongs to B\A and z=x*y. So we have x=z/y belongs to B, but x can not belong to B because x is the difference A\B. So we have a contradiction when we can not choose two elements, one of which could be the difference between A and B, and other one - the difference between B and A, So one of these subgroups can not have a difference from the second one, it can only fully belong to the other group. So one of two groups A and B is the main group, R, and another - its subgroup, but firstly was said that proper ideal of a ring is a subgroup of this ring which is not equal to the ring itself. So the ring can not be expressed as a union of two proper ideals. The first statement is proved.

And the proof of the second statement follows from the above with little difference - in this case we also have a subgroup C, z=x*y can belong to it instead of A or B. It is a third case which we could not consider in the first part not having a third subgroup C. It gives us the opportunity of existing of the ring which is union of A, B and C because z does not belong to A or B (it belongs to C), so we can avoid the situation which is given above which obliges us to have x and y both belonging to A or B. Compare this case with the first one when x belongs to A and z=x*y belongs to A. It makes y belong to A because y=z/x. And the second case makes x belong to B because x=z/y. So we have both x and y in A or B. But if z belongs to C, neither to A nor to B, the restriction is lifted and we can have 3 independent subgroups A, B, C forming a ring (group) R. The example of this case can be Klein quadruple group representable as a union of three of its isomorphic subgroups , none of which lies entirely in the other {1, a, b, ab}