Connection between ideals in a ring and ring with associated unit

Let $(A,+,\cdot)$ be a ring. In the set $A\times \mathbb{Z}$ we define: $$(a,k)+'(b,l)=(a+b,k+l),$$ $$(a,k)\cdot ' (b,l):=(ab+la+kb,kl)$$ for $a,b \in A$, $k,l \in \mathbb{Z}$. Then $(A\times \mathbb{Z}, +',\cdot')$ is the unitary ring.

Is there connection between ideals in $A$ and ideals in $A\times \mathbb{Z}$ ?

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What is $P$? Is that supposed to be $A\times \mathbb Z$? –  Thomas Andrews Feb 24 '12 at 22:05
Sorry. It should be $A\times Z$. –  L.T Feb 24 '12 at 22:20

1 Answer

First, $A\times\mathbb{Z}$ is not a good notation for the ring (known as the Dorroh extension of $A$), because $A\times\mathbb{Z}$ has a natural ring structure which is not the one we are using here. So it's best to call it something else. It's common to use $A^1$.

Second: There is a canonical embedding $\varphi\colon A\to A^1$ given by $\varphi(a)=(a,0)$. Under this embedding, you can verify rather easily that $\varphi(I) = \{(x,0)\mid x\in I\}$ is a left (resp. right, two-sided) ideal of $A^1$ whenever $I$ is a left (resp. right, two-sided) ideal of $A$.

If $J$ is a left (resp. right, two-sided) ideal of $A^1$, then both the intersection with $\varphi(A)$ and the projection onto $A$ are left (resp. right, two-sided) ideals of $A$.

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