We can think differently about networks and subnets using modular arithmetic & ring theory. Should we? Probably not.

[2001 : d b 8 ::] (\mod 2^{64})

IPv6 = Z / 2^{128} Z

What if networks are just rings?

IP addresses look like boring dotted decimals: 192.168.1.1, 10.0.0.42, 8.8.8.8. But under the hood, networking is pure binary. And binary means math. And math means we can do cursed stuff.

In fact:

IPv4 space = Z / 2^{32} Z

The space of IPv4 addresses is just the commutative residue class ring of integers mod $2^{32}$ .

NOW, is that useful?

The answer might shock you...
(it's "no")

IP addresses as a ring

Each IPv4 address = an element of $Z / 2^{32} Z$ .
Addition = add two IPs, wrap around at $2^{32}$ .
Multiplication = defined mod $2^{32}$ as well (not used in networking, but the ring structure exists).

So 0.0.0.0 = 0, 255.255.255.255 = $2^{32} - 1$ , and your home router at 192.168.0.1 is just 3232235521 in $Z / 2^{32} Z$ .

Subnets/Masks as ideals

CIDR blocks like 192.168.0.0/24 are usually taught as "keep the first 24 bits fixed, vary the last 8."

But in ring language:

a /n subnet = a + (2^{32 - n})

That is: a coset of the ideal $(2^{32 - n}) \subseteq Z / 2^{32} Z$ .

/24 → cosets mod $2^{8}$ = groups of 256 consecutive addresses.
/16 → cosets mod $2^{16}$ = groups of 65,536.
/32 → cosets mod $2^{0} = 1$ = just one address.

So...

Every subnet is just congruence classes in the ring.

Notation

Instead of 192.168.0.0/24, we could write:

[3232235520] (\mod 256)

Meaning: all addresses congruent to 192.168.0.0 modulo 256.

Clean, right? Hmm... Well, not really.
But this does scale for IPv6 (128 bits), it could be nice to write:

[2001 : d b 8 ::] (\mod 2^{64})

instead of a wall of colons and hex digits with /64.

It could be much simpler to describe "all addresses congruent mod $2^{64}$ " than to eyeball giant hex blocks.

Some interesting connections

If IPv4 is $Z / 2^{32} Z$ and IPv6 is $Z / 2^{128} Z$ , then subnetting, addressing, and even "address arithmetic" are living in a ring. That raises the question: how do deeper ring-theoretic concepts reflect on networking?

Partitioning:
Ideals in $Z / 2^{32} Z$ partition the ring into cosets. Subnets partition address space into networks. Ideals = subnets, Cosets = host groups, Lattice of ideals = the hierarchy of subnetting.
Local ring:
$Z / 2^{32} Z$ has a unique maximal ideal $(2)$ . In networking terms, "everything breaks down into powers of 2". Exactly how subnetting works.
Zero divisors:
Not every element has a multiplicative inverse (because $2^{32}$ isn't prime). Networking analogy (not exact, but similar): some addresses don't behave like regular hosts (network and broadcast addresses "collapse" the same way).
Addition wraps around:
Just like incrementing 255.255.255.255 brings you back to 0.0.0.0.
Uniform notation:
a mod 2^k instead of dotted-decimal + /n.
Additive identity (0):
In the ring, $0$ is the neutral element. In networking, that's the all-zero address (0.0.0.0 or ::). It plays a "special" role: the unspecified address or "this host" in routing.
Prime ideals:
There are no prime ideals in $Z / 2^{n} Z$ beyond $(2)$ . That reflects the rigid structure of subnetting: everything reduces to powers of two. No alternative partition sizes exist (you can't subnet in blocks of 3 or 5). This is why networks always split cleanly into halves, quarters, eighths... never thirds.
Polynomials over the ring:
Consider $(Z / 2^{n} Z) [x]$ . You can think of this as "networks with an extra dimension." For example, addresses + time, or addresses + port numbers. In fact, transport protocols already do something like this: a socket is IP × port → essentially a polynomial ring structure on top of the base ring. Here, IP × port behaves like adding another dimension.

Ring homomorphisms

From IPv6 to IPv4:

φ : Z / 2^{128} Z \to Z / 2^{32} Z, φ (x) = x mod 2^{32}

This is exactly what "IPv4-mapped IPv6 addresses" do (::ffff:192.168.0.1). It's a homomorphism that forgets 96 high-order bits.

From subnets to smaller subnets: restricting a coset modulo $2^{k}$ to modulo $2^{m}$ ( $m < k$ ) is a homomorphism that corresponds to refining a network into smaller blocks.

Take IPv6 address space $Z / 2^{128} Z$ . The natural ring homomorphism

φ (x) = x mod 2^{64}

projects every IPv6 address onto its subnet prefix (/64). This maps the 128-bit space onto the 64-bit "network" space. It's exactly what routers care about: the higher-order bits decide the routing table, the lower-order bits are thrown away. That's algebra and networking aligning perfectly.

Chinese Remainder Theorem:

Normally, the CRT lets you split $Z / n Z$ into a product of smaller rings corresponding to the prime-power factors of $n$ . More specifically, if

n = p_{1}^{k_{1}} p_{2}^{k_{2}} \dots p_{m}^{k_{m}}

is the prime factorization of $n$ , then

Z / n Z ≅ Z / p_{1}^{k_{1}} Z \times Z / p_{2}^{k_{2}} Z \times \dots \times Z / p_{m}^{k_{m}} Z .

But here, $2^{n}$ has only one prime factor. That's why IP space only has the tidy binary subnetting structure. If addresses had modulus $6$ , subnetting would involve mod 2 and mod 3 simultaneously.

Lattice of ideals ↔ subnet hierarchy

In commutative algebra:

The set of ideals of a ring, ordered by inclusion, forms a lattice (a partially ordered set where any two have a unique meet and join).
In $Z / 2^{n} Z$ , the ideals are exactly $(2^{k})$ for $0 \leq k \leq n$ .
They're nested:
$(1) \supset (2) \supset (2^{2}) \supset \dots \supset (2^{n}) \supset (0) .$

In networking:

/0 (the whole internet) contains /1 networks.
/1 contains /2.
/2 contains /3.
... all the way down to /32 (a single host).

That's exactly the same chain structure as the ideals. Bigger subnet (smaller prefix length) = bigger ideal. Smaller subnet (longer prefix length) = smaller ideal.
So:

In algebra: ideals $(2^{k})$ form a descending chain.
In networking: subnets /k form a descending chain of refinements.

Well

That's not just an analogy. It's literally the same lattice structure.

TL;DR: In other words

IPv4 = $Z / 2^{32} Z$ .
A /n subnet = a coset of the ideal $(2^{32 - n})$
- Ideals = subnets.
- Cosets = host groups.
- Lattice of ideals = the hierarchy of subnetting.
Quotients = collapsing hosts into a network.
Homomorphisms = projection from big address spaces to smaller, just like prefix matching.
CRT doesn't add extra structure here (because powers of 2), which explains why IP networks are powers-of-two only.

So IMO Networking is number theory with better marketing.

What if Networks are Rings?

What if networks are just rings? ​

IP addresses as a ring ​

Subnets/Masks as ideals ​

Notation ​

Some interesting connections ​

Ring homomorphisms ​

Chinese Remainder Theorem: ​

Lattice of ideals ↔ subnet hierarchy ​

TL;DR: In other words ​