Department of Mathematical Sciences Instructor: Daiva Pucinskaite Modern Algebra April 11, 2017

Homework (Rings)

Definition. A set R with two operations ⊕, and is called a ring if 1 h R, ⊕i is an abelian group. (i) The identity element in R w.r.t. ⊕ is called the zero element and is written 0. (ii) The inverse element of a in R w.r.t. ⊕ is called the negative element of a, and will denoted by −a. 2 is associative, i.e. (a b) c = a (b c) for all a, b, c ∈ R. 3 a (b⊕c) = (a b)⊕(a c) and (b⊕c) a = (b a)⊕(c a) for all a, b, c ∈ R, in this case we say ”multiplication is distributive over addition.” Definition. Let hR, ⊕, i be a ring. (•) If a b = b a for all a, b ∈ R, then R is called a commutative ring. (•) If there is in R a neutral element w.r.t. (i.e. there is x ∈ R such that x a = a x = a for all a ∈ R) then this element is called an unity of R, and is denoted by the symbol 1. (•) If R has an unity element, then R is called a ring with unity. (•) If R a ring with an unity 1, and for an element a in R there is an element x in R such that a x = 1 as well as x a = 1 (in other words, an element a in R has a multiplicative inverse), then a is called invertible. (•) If R is called a field if (i) R is commutative, (ii) R a ring with unity, (iii) each element in R, exempt 0, the zero element (i.e. exempt the identity element in R w.r.t. ⊕), is invertible. (•) An element a in R is called a divisor of zero if (i) a 6= 0 (i.e. a is not the identity element in R w.r.t. ⊕), (ii) there exists an non-zero element b in R (i.e. b 6= 0) such that a b = 0. (•) R is called an integral domain if (i) R is commutative, (ii) R is a ring with unity, (iii) R has no divisors of zero.

I. Consider the set Z with conventional addition and multiplication, here ⊕ = +, and = ·. 1. Show that Z = hZ, +, ·i is a ring. 2. Show that Z is commutative. 3. Show that Z is ring with unity. 4. Show that Z has precisely two invertible elements. 5. Conclude from 4. that Z is not a field. 6. Show that Z has no divisors of zero. 7. Conclude from 1.,2.,3., and 6. that Z is an integral domain. II. Consider the set R with conventional addition and multiplication, here ⊕ = +, and = ·. 1. Show that R = hR, +, ·i is a ring. 2. Show that R is commutative. 3. Show that R is ring with unity. 4. Show that each non-zero real number is invertible. 5. Conclude from 1.,2.,3., and 4. that R is a field. 6. Show that R has no divisors of zero. 7. Conclude from 1.,2.,3., and 6. that R is an integral domain.

III. Consider the set Z6 = {0, 1, 2, 3, 4, 5} with ⊕ = +, and = · defined as follows: Let i, j ∈ Z6, then i · j = i+i+ ··· +i | {z } j−times

1. Show that Z6 = hZ6, +, ·i is a ring. 2. Show that Z6 is commutative. 3. Show that Z6 is ring with unity. 4. Show that 1 and 5 are the only invertible elements in Z6. 5. Conclude from 4. that Z6 is not a field. 6. Show that 2, 3 and 4 are the divisors of zero.

7. Conclude from 6. that Z6 is not an integral domain.

IV. Consider the set Z5 = {0, 1, 2, 3, 4} with ⊕ = +, and = · defined as follows: Let i, j ∈ Z5, then

i · j = i+i+ ··· +i | {z } j−times

1. Show that Z5 = hZ5, +, ·i is a ring. 2. Show that Z5 is commutative. 3. Show that Z5 is ring with unity. 4. Show that each non-zero element in Z5 is invertible. 5. Conclude from 1.,2.,3., and 4. that Z5 is a field. 6. Show that Z5 has no divisors of zero. 7. Conclude from 1.,2.,3., and 6. that Z5 is an integral domain.

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