Ring with Unity

Definition and Basic Properties

A ring with unity (also known as a unital ring or ring with identity) is a fundamental concept in abstract algebra. It is a set equipped with two binary operations, typically called addition and multiplication, that satisfies the properties of a ring and also contains a multiplicative identity element, often denoted as 1. Formally, a ring \( R \) with unity is a set \( R \) together with two operations \( + \) and \( \cdot \) such that:

1. \( (R, +) \) is an abelian group. 2. \( (R, \cdot) \) is a monoid with identity element 1. 3. Multiplication is distributive over addition, i.e., for all \( a, b, c \in R \):

  - \( a \cdot (b + c) = (a \cdot b) + (a \cdot c) \)
  - \( (a + b) \cdot c = (a \cdot c) + (b \cdot c) \)

The presence of a multiplicative identity distinguishes rings with unity from more general rings that may lack such an element.

Examples of Rings with Unity

Integers

The set of all integers \( \mathbb{Z} \) with the usual addition and multiplication is a ring with unity. The multiplicative identity in this case is the number 1.

Matrices

The set of all \( n \times n \) matrices over a ring \( R \), denoted by \( M_n(R) \), forms a ring with unity. The multiplicative identity is the identity matrix \( I_n \), which has 1's on the diagonal and 0's elsewhere.

Polynomial Rings

The set of all polynomials with coefficients in a ring \( R \), denoted by \( R[x] \), is a ring with unity. The multiplicative identity is the constant polynomial 1.

Image of polynomial equations written on a chalkboard.

Image of polynomial equations written on a chalkboard.

Properties and Theorems

Units and Invertibility

In a ring with unity, an element \( a \in R \) is called a unit if there exists an element \( b \in R \) such that \( a \cdot b = b \cdot a = 1 \). The set of all units in \( R \) forms a group under multiplication, known as the unit group of \( R \).

Homomorphisms

A ring homomorphism between two rings with unity \( R \) and \( S \) is a function \( f: R \to S \) that preserves the ring operations and maps the unity of \( R \) to the unity of \( S \). That is, for all \( a, b \in R \): - \( f(a + b) = f(a) + f(b) \) - \( f(a \cdot b) = f(a) \cdot f(b) \) - \( f(1_R) = 1_S \)

Ideals and Quotient Rings

An ideal \( I \) of a ring \( R \) is a subset of \( R \) that is closed under addition and under multiplication by any element of \( R \). If \( R \) is a ring with unity, the quotient ring \( R/I \) inherits a unity from \( R \), provided that \( I \) is a proper ideal.

Special Types of Rings with Unity

Commutative Rings

A ring with unity is called a commutative ring if the multiplication operation is commutative, i.e., \( a \cdot b = b \cdot a \) for all \( a, b \in R \).

Division Rings

A division ring (or skew field) is a ring with unity in which every non-zero element is a unit. If the ring is also commutative, it is called a field.

Integral Domains

An integral domain is a commutative ring with unity that has no zero divisors. This means that if \( a \cdot b = 0 \) for \( a, b \in R \), then either \( a = 0 \) or \( b = 0 \).

Applications

Rings with unity are ubiquitous in mathematics and its applications. They appear in various branches such as number theory, algebraic geometry, and functional analysis. For instance, the ring of continuous functions on a topological space, with pointwise addition and multiplication, is a ring with unity.

See Also

• Field
• Integral Domain
• Unit
• Ring Homomorphism
• Polynomial Ring