MAT3702
Assignment 2
Due: 15 August 2025
, MAT 3702 Assignment 2 2025 Solutions
Due Date: 15 August 2025
Question 1(i)
Problem Statement: Let R be a ring with a, b, c ∈ R and 0R ∈ R be the zero element
of R. If a + b = a + c, then show that b = c.
Step 1: Understand the given equation and ring properties
Given a + b = a + c in a ring R. A ring has commutative and associative addition,
and every element has an additive inverse. The zero element 0R satisfies x+0R = x.
Our goal is to show b = c.
Step 2: Subtract a from both sides
To isolate b and c, add the additive inverse −a to both sides:
(a + b) + (−a) = (a + c) + (−a)
Step 3: Apply associativity of addition
Using associativity:
a + (b + (−a)) = a + (c + (−a))
Step 4: Use the additive inverse property
Since a + (−a) = 0R , consider adding −a correctly:
(−a) + (a + b) = (−a) + (a + c)
Regroup:
((−a) + a) + b = ((−a) + a) + c
Since (−a) + a = 0R :
0R + b = 0R + c =⇒ b = c
Step 5: Verify the simplification
The commutative and associative properties allow cancellation of a, confirming:
b=c
Final Answer: b = c
1
Assignment 2
Due: 15 August 2025
, MAT 3702 Assignment 2 2025 Solutions
Due Date: 15 August 2025
Question 1(i)
Problem Statement: Let R be a ring with a, b, c ∈ R and 0R ∈ R be the zero element
of R. If a + b = a + c, then show that b = c.
Step 1: Understand the given equation and ring properties
Given a + b = a + c in a ring R. A ring has commutative and associative addition,
and every element has an additive inverse. The zero element 0R satisfies x+0R = x.
Our goal is to show b = c.
Step 2: Subtract a from both sides
To isolate b and c, add the additive inverse −a to both sides:
(a + b) + (−a) = (a + c) + (−a)
Step 3: Apply associativity of addition
Using associativity:
a + (b + (−a)) = a + (c + (−a))
Step 4: Use the additive inverse property
Since a + (−a) = 0R , consider adding −a correctly:
(−a) + (a + b) = (−a) + (a + c)
Regroup:
((−a) + a) + b = ((−a) + a) + c
Since (−a) + a = 0R :
0R + b = 0R + c =⇒ b = c
Step 5: Verify the simplification
The commutative and associative properties allow cancellation of a, confirming:
b=c
Final Answer: b = c
1
