Notes on algebra, probability theory, and linear algebra
A ring is a set $R$ with two binary operations, usually called addition and multiplication, such that:
\(a(b+c)=ab+ac,\) and
\((a+b)c=ac+bc.\) Some authors require a multiplicative identity $1$; some do not. These notes focus on the conceptual role of the two operations and distributivity.
A ring is a structure where objects can be both added and multiplied, and multiplication respects addition.
Addition represents combining contributions.
Multiplication represents interaction, composition, scaling, or product formation.
Distributivity is the bridge between them.
One operation gives one kind of composition.
Two operations allow two different modes:
| Operation | Typical meaning |
|---|---|
| Addition | accumulation, superposition, sum of contributions |
| Multiplication | interaction, scaling, composition, product |
Many mathematical worlds naturally have both:
The key law is
\(a(b+c)=ab+ac.\) It says:
If something is decomposed as a sum of parts, then multiplying it by $a$ can be done part by part.
Equivalently:
Multiplication is compatible with decomposition into additive contributions.
This is the central meaning of distributivity.
If $b+c$ is a whole made of two contributions, then $a(b+c)$ means letting $a$ interact with the whole.
Distributivity says this is the same as:
So distributivity is the principle:
The whole can be computed from its parts.
A rectangle of height $a$ and width $b+c$ has area
\(a(b+c).\) If we cut it into two rectangles of widths $b$ and $c$, the areas are
\(ab \quad\text{and}\quad ac.\) The total area is
\(ab+ac.\) So
\(a(b+c)=ab+ac.\) Here distributivity says: area of the whole is the sum of the areas of the parts.
For fixed $a$, the map
\(x\mapsto ax\) preserves addition:
\(a(x+y)=ax+ay.\) So multiplication by $a$ is a homomorphism of the additive group.
In other words:
Multiplication acts linearly on the additive world.
This is why rings connect so strongly with linear algebra.
If addition and multiplication are not connected by distributivity, then they are almost two unrelated operations on the same set.
We lose:
Without distributivity, expressions like
\(a(b+c) \quad\text{and}\quad ab+ac\) need not have any relationship.
\(\mathbb Z\) with ordinary addition and multiplication is the prototypical ring.
\(K[x]\) is a ring. Addition combines coefficients, multiplication combines powers and coefficients.
\(M_n(K)\) is a ring under matrix addition and multiplication.
Multiplication is generally non-commutative, but it is associative and distributive over addition.
Functions $X\to K$ form a ring under pointwise operations:
\((f+g)(x)=f(x)+g(x),\) \((fg)(x)=f(x)g(x).\)
Distributivity is not the only possible compatibility law between two operations.
Other structures use different laws.
One can simply put two operations on a set without requiring a relation. This is possible but usually too weak to support a rich theory.
Some structures require only one distributive law, such as
\(a(b+c)=ab+ac,\) but not necessarily
\((a+b)c=ac+bc.\) These lead to structures such as near-rings.
Lattices use operations $\vee$ and $\wedge$ with absorption laws:
\(a\vee(a\wedge b)=a,\) \(a\wedge(a\vee b)=a.\) This models union/intersection-like behavior rather than addition/multiplication-like behavior.
Boolean algebras have two operations that distribute over each other, reflecting logical AND/OR or intersection/union.
Rings are where algebra becomes the study of expressions involving both addition and multiplication:
\(a^2+ab+b^2, \quad (a+b)(c+d), \quad x^3-2x+5.\) They are the natural setting for arithmetic, polynomials, matrices, and many algebraic constructions.
The next step is to ask when multiplication is as well-behaved as possible. That leads to fields.