Does there exists a Ring with unity other than Z4 such that for all non-zero non-unit elements of ring, 2x=0 and x⋅x=0 Let R be a ring with unity and x+x=0,x⋅x=0 for all non-zero non unit element x of R. Z4 is one of the example of that ring. Does there exists any other ring of this type ? If no then how can we prove that.

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Does there exists a Ring with unity other than Z4 such that for all non-zero non-unit elements of ring, 2x=0 and x⋅x=0  Let R be a ring with unity and x+x=0,x⋅x=0 for all non-zero non unit element x of R.   Z4 is one of the example of that ring. Does there exists any other ring of this type ? If no then how can we prove that.

Gerald Lopez · Accepted Answer

There are a huge number of examples furnished by a construction known as the trivial extension.  If you take an elementary abelian 2-group G whatsoever, and write F2 to mean the field of 2 elements, then R=F2×G is an example. Firstly the ring has characteristic 2 so the x+x=0 condition is trivially satisfied by all elements, and secondly because 1+G is the set of units and the complement G is a subset that squares to zero.  Another observation is that if R is assumed to be commutative, x2=0 for all nonunits implies that the ring is local, and that its maximal ideal squares to zero. The other condition then implies that the maximal ideal is a rng of characteristic 2.

Jenny Bolton · Accepted Answer

Any field satisfies x⋅x=0 and 2⋅x=0 for any non-unit element x. Another example would be the quotient of the polynomial ring (Z2Z)[Xi∣i∈I] modulo the ideal ⟨Xi2∣i∈I⟩.

Does there exists a Ring with unity other than Z_{4} such that for all non-zero non-unit eleme

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