Let \[ \mathbb Z[i]=\{a+bi:a,b\in\mathbb Z\}. \] List all subrings of \(\mathbb Z[i]\), with each subring appearing exactly once.
Solution
For every integer \(n\ge1\), define \[ R_n=\ZZ+n\ZZ i = \{a+nbi:a,b\in\ZZ\}. \]
1. The sets\[ \ZZ,\qquad R_1,\qquad R_2,\qquad\ldots \]are subrings of \(\ZZ[i]\).
The set \(\ZZ\) is a subring. Fix \(n\ge1\). Clearly \(0,1\in R_n\), and \(R_n\) is closed under additive inverses and addition. If \[ x=a+nbi, \qquad y=c+ndi \] belong to \(R_n\), then \[ xy = (ac-n^2bd)+n(ad+bc)i \in R_n. \] Thus \(R_n\) is also a subring.
2. Every subring \(R\subseteq\ZZ[i]\) contains \(\ZZ\).
Under the convention for rings and subrings used here, \(R\) contains the same identity \(1\). Hence it contains every integer multiple of \(1\), so \[ \ZZ\subseteq R. \]
3. Additive subgroups of \(\ZZ[i]\) containing \(\ZZ\) are inbijection with subgroups of\[ \ZZ[i]/\ZZ\cong\ZZ. \]
The correspondence theorem for groups sends an additive subgroup \(H\) with \[ \ZZ\subseteq H\subseteq\ZZ[i] \] to \(H/\ZZ\). The quotient is isomorphic to \(\ZZ\) via \[ (a+bi)+\ZZ\longmapsto b. \]
4. The inverse images of the subgroups of \(\ZZ\) under the map instep 3 are precisely\[ \ZZ \]and\[ R_n=\ZZ+n\ZZ i \qquad(n\ge1). \]
Every subgroup of the additive group \(\ZZ\) is either \(\{0\}\) or \(n\ZZ\) for a unique integer \(n\ge1\). The inverse image of \(\{0\}\) is \(\ZZ\). The inverse image of \(n\ZZ\) consists exactly of the Gaussian integers whose imaginary coefficient is divisible by \(n\), namely \(R_n\).
5. Every subring of \(\ZZ[i]\) is exactly one member of the list instep
1.
Let \(R\) be a subring. By step 2, it is an additive subgroup containing \(\ZZ\). Steps <1>3–<1>4 therefore show that \[ R=\ZZ \] or \[ R=R_n \] for some \(n\ge1\).
The list has no repetitions: \(\ZZ\) has only elements with imaginary coefficient \(0\), while \(R_n\) contains \(ni\); and if \(R_m=R_n\), then their imaginary-coordinate subgroups satisfy \[ m\ZZ=n\ZZ, \] which forces \(m=n\) for positive \(m,n\).
6. Thus the complete classification is\[ \boxed{ \ZZ,\quad \ZZ+i\ZZ,\quad \ZZ+2i\ZZ,\quad \ZZ+3i\ZZ,\quad\ldots }. \]
Step 1 shows that every listed set is a subring, and step 5 shows that every subring occurs exactly once.
7. Q.E.D.
Step 6 answers the problem.