Chapter Two Ring Theory 2-1 Definitions and examples Definition 2.1.1 A ring R is a set together with two binary operation and (called addition and multiplication defined on R) if satisfying the following axioms: (1) is an abelian group, (2) For all implies (3) . is associative: (4) the distributive law hold in R: , and Example. , are ring. Definition 2.1.2. The ring is commutative if multiplication is commutative. Definition 2.1.3. The ring is said to be ring with identity if Example: are commutative ring with identity. Definition 2.1.4. Let be a ring with identity .we say that an element has an inverse a unit element if it has the multiplicative inverse ( i.e We denoted the set of all unit elements in by Theorem 2.1.5. Let be a ring with identity. Then is a group. Proof. Since , then is a non-empty set. Now we prove that the axioms of group are satisfies: 1- let that is each of has inverse multiplication. Hence and This implies that is invers of and . Hence the set is closed under multiplication. 2- associative law are holds because is ring. 3- is identity element. 4- If , then is group. Example.(1) In we see and is an abelian group. (2) Let be a non-empty set. If is a power set of , then show that Is a commutative ring with identity. (3) Let be the square matrix of . Show that a ring with identity. Definition 2.1.6. Let be a ring. For all and for all integer define and define If R with identity, then If R with identity and a has a multiplicative inverse, then Theorem 2.1.7. Let be a ring, for and arbitrary integers n the following hold: 1- 2- 3- Theorem 2.1.8. Let be a ring and be a zero element. The for all the following hold: 1- 2- 3- 4- Proof. 1- since Hence . By cancellation law we get the result. Similarly we can proof . 2-H.w 3-By (2) we get 4-H.w Corollary 2.1.9. Let be a ring with identity such that . Then the element and are distinct. Proof. Since , there exists some nonzero element Now if , It follow that , which is contradiction. Corollary 2.1.10. Let be a ring with identity such that . Then for all the following are hold: 1- and 2- Definition 2.1.11. Let be a ring and let be a non empty subset of . If is itself a ring, then is said to a subring of Remark. Every ring has two trivial subring; for, if 0 denote the zero element of the ring , then both and the ring itself are subrings of Definition 2.1.12. Let be a ring and . Then is a subring of if and only if 1- – ( closed under differences) 2- ( closed under multiplication) Examples. 1- is a subring of and 2- is a subring of . 3- Let R denote the set of all functions The sum and the product of two function are defined by Suppose is the commutative ring of function of above. Define Definition 2.1.13. The center of a ring denoted by , is the set . Remark. If is comuutaive, then Theorem 2.1.14. Let be a ring. Then is a subring of . Proof. Since , then , hence Let . To prove that For all , then Therefore and Therefore hence is a subring of . 2-2 Some type of rings. Definition 2.2.1. A ring is said to hane divisors of zero if there exist nonzero elements such that the product . Definition 2.2.2. An integral domain is a commutative ring with identity which does not have divisors of zero. Examples. are integral domain but is not integral domain. Theorem 2.2.3. Let be a commutative ring with identity. Then is an integral domain if and only if the cancellation law holds for multiplication. Proof. We suppose that R is an integral domain . Let such that and . Hence Conversely, suppose that the cancellation law holds and . If the element , then by Theorem 2.1.6 we have , hence , consequently . That is has no divisors of zero and commutative with identity, we get is an integral domain. Corollary 2.2.4. Let be an integral domain. Then the only solution of the equation are and . Proof. Clearly 0 is the solution of the equation Now , if , since and , hence by cancellation law we get Definition 2.2.5. A ring (R, +, .) is said to be a division ring if it is a ring with identity in which every nonzero element has a multiplicative inverse. Definition 2.2.6. A field is a commutative ring with identity in which each nonzero element has an inverse under multiplication. Examples: 1- are field. 2- (Zn ,+n ,.n) is a field if and only if n is a prime number. 3- is an integral domain but not a field. Theorem2.2.7. Every field is an integral domain. Proof. Let ( be a field. Then R is a commutative ring with identity. Let . Since R is a field, then the element a has an inverse.. The hypothesis a.b =0 yields That is contains no divisors of zero. Hence is an integral domain. Theorem2.2.8. Any finite integral domain is a field. Proof. Let ( be an integral domain contains n distinct elements say . Let be any element of , consider the elements These products are all distinct because If but which is contradiction to are all distinct. Since , then for some and has multiplicative inverse and . That is is a field. Theorem 2.2.9. The ring of integers modulo n is a field if and only if n is a prime number. Proof. Suppose that is a field. To prove that is a prime number. If is not prime, then where It follows . Since This means that the system is not an integral domain and hence not a field. Conversely suppose that n is a prime number. To prove that (Zn ,+n , .n) is a field, enough to show that is an integral domain. Let and Hence has no divisors of zero, that is is an integral domain. Definition 2.2.10. Let be an arbitrary ring. If there exists a positive integer such that for all , then the least positive integer with this property is called the characteristic of the ring. If no such positive integer exists, then we say (R, +, .) has characteristic zero. Definition 2.2.11. : Let be a ring with identity. Then has characteristic if and only if is the least positive integer for which . Proof: If the ring is of characteristic , it follows that . Were , where then for every This mean The characteristic of is less than , which is contradiction. Conversely, Let be the least positive integer in which Let Then has characteristic . Corollary 2.2.12. The characteristic of an integral domain is either zero or a prime. Proof. Let be a positive characteristic n and assume that is not a prime Then n can be written as with . By Theorem 2.2.11 we have Since by hypothesis is without zero divisors, then either But this contradicts the choice of as the least positive integer such that Hence the characteristic of must be prime. Example. Show that the characteristic of the ring is equal two. Since is the zero element of the ring Now for all , then – – . From the definition of characteristic, then the characteristic of is 2. 2-3 Ideals and Quotient rings. Definition 2.3.1. A subring of the ring is an ideal of if and only if and imply both and Definition 2.3.2. Let be a ring and a non-empty subset of . Then is an ideal of if and only if (1) imply , (2) imply both Remark. In a commutative ring , every right ideal is left ideal. Examples. 1) The subring is an ideal of 2) The trivial subrings of the ring ( are both ideals. Any ideal different from is called proper ideal. 3) In the ring for a fixed integer . Then is an ideal of because where 4) is not ideal of but is a subring of Since and , then . Then is not ideal of 5) Let be the square matrix ring over the field of real number. Then is not an ideal. Definition 2.3.3. A ring which contains no ideals except trivial ideals is said to be a simple ring. Definition 2.3.4. Let be a commutative ring with identity. An ideal is called a principal ideal of the ring if generated by a single element and denoted by . Example. In the ring ( the ideal is a principal ideal generated by 2 and is a principal ideal generated by 3. Theorem 2.3.5. If is an ideal of the ring then for some nonnegative integer . Proof. If then the theorem is true. Suppose then that that is there exists Since I is an ideal, then – so contains positive integers. Let be the least positive integer in . We claim Since and is an ideal of then , for all , that is On the other hand, any integer . By division Algorithm there exists such that , where . Since are members of , it follows that . Our be a least integer implies , and consequently Therefore . Definition 2.3.6. Let be a commutative ring with identity. A ring is called a principal ideal ring if every ideal is principal. Theorem 2.3.7. Let be a ring with identity element and be an ideal of containing identity element .Then .