Here is a clear, simple, and exam-oriented explanation of Subrings in Abstract Algebra / Discrete Structures.

---

⭐ Subrings – Introduction

A subring is a subset of a ring that is itself a ring with the same operations (addition and multiplication) as the larger ring.

Simple idea:

A subring is like a “smaller ring inside a bigger ring.”

---

⭐ Formal Definition

Let ((R, +, \cdot)) be a ring.
A non-empty subset (S \subseteq R) is a subring of R if:

1. S is closed under subtraction:
   [
   a, b \in S \Rightarrow a – b \in S
   ]
2. S is closed under multiplication:
   [
   a, b \in S \Rightarrow ab \in S
   ]

If these two conditions hold, then S automatically becomes a ring under the same operations.

These two conditions are known as the Subring Test.

---

⭐ Why “Subtract”?

Because requiring:

• closed under addition
• closed under additive inverses

is equivalent to closure under subtraction.

Subtraction ensures the subset forms an additive subgroup of the ring.

---

⭐ Subring Test (Exam Point)

A non-empty subset (S) of a ring (R) is a subring iff:

1. a − b ∈ S for all a, b ∈ S
2. ab ∈ S for all a, b ∈ S

This is the most commonly used test in exams.

---

⭐ Examples of Subrings

✔ Example 1: Even integers (2ℤ) ⊆ ℤ

Let R = ℤ (integers)
Let S = 2ℤ = { …, -4, -2, 0, 2, 4, … }

Check:

• Closed under subtraction:
  4 − 2 = 2 ∈ S
  -6 − 2 = -8 ∈ S
• Closed under multiplication:
  2 × 4 = 8 ∈ S
  -2 × 6 = -12 ∈ S

So 2ℤ is a subring of ℤ.

---

✔ Example 2: nℤ is a subring of ℤ

For any integer n,
nℤ = {0, ±n, ±2n, ±3n, …}
is a subring of ℤ.

---

✔ Example 3: Z is a subring of Q, R, and C

Integers ℤ are contained in:

• rationals ℚ
• reals ℝ
• complex numbers ℂ

and closed under subtraction & multiplication.
So ℤ is a subring of each.

---

✔ Example 4: Set of 2×2 matrices with integer entries

If R = all 2×2 real matrices
S = all 2×2 integer matrices

Then S is a subring of R.

---

⭐ Examples That Are NOT Subrings

✘ Natural numbers ℕ (not a subring of ℤ)

ℕ = {1,2,3,…}
Not closed under subtraction:

3 − 5 = −2 ∉ ℕ

So ℕ is not a subring of ℤ.

---

✘ Odd integers are NOT a subring

Odd numbers: {…, -3, -1, 1, 3, …}

1 − 1 = 0 ∉ S
So it fails closure.

---

⭐ Properties of Subrings

If S is a subring of R:

• S must contain 0 (additive identity)
• S must contain a − b for all a, b ∈ S
• S must contain ab for all a, b ∈ S
• S may not contain identity 1 of R
  (subrings need not have the same 1)

---

⭐ Difference Between Ring and Subring

Ring	Subring
Complete algebraic structure	Smaller ring inside a ring
Has two operations	Inherits operations
May contain identity 1	Subring may or may not contain 1
Example: ℤ	Subring: 2ℤ

---

⭐ Quick Exam Answer

Definition:
A subset S of a ring R is a subring if (S, +, ·) is itself a ring with the same operations as R.

Subring Test:
A non-empty subset S is a subring of R if:

1. ( a – b \in S \quad \forall a, b \in S )
2. ( ab \in S \quad \forall a, b \in S )

Examples:

• 2ℤ is a subring of ℤ
• ℤ is a subring of ℚ, ℝ, ℂ
• Integer matrices form a subring of real matrices

---