 Hello and welcome to the session on the topic Algebraic Systems with two binary operations under the course Discrete Mathematical Structures at second year of Information Technology Engineering semester one. At the end of this session students will be able to demonstrate ring, ring homomorphism, sub ring and field and its properties. These are the contents we are going to cover ring, ring homomorphism, sub ring, field with its definition and examples. To start with we will see the definition for an algebraic structure termed as a ring. An algebraic system s, plus, dot is called a ring if the binary operation plus and dot on s satisfy the following three properties. Number one, s, plus is an abelian group, number two, s, dot is a semi group and number three the operation dot is distributive over plus that is for any a, b, c which belongs to s we get a dot b plus c is equal to a dot b plus a dot c and b plus c dot a is equal to b dot a plus c dot a. If you recollect we have already defined the various other definitions such as semi groups, monoids, groups, abelian group and so on. Using those definitions now we come out with a new algebraic system called as ring and this is defined on the basis of three properties. So let us discuss one by one. Number one condition says s, plus must be an abelian group. Now when we consider the algebraic system s, plus, dot we observe that unlike the earlier definitions it contains an algebraic system with two binary operations and which are these two operations namely plus and dot. Now based upon the context each of plus and dot may have different meanings like plus may be addition and dot may be multiplication. So to start with this algebraic system which contains the set s along with two binary operations. Number one we define the algebraic system with single binary operation. First of all s, plus must be an abelian group. Number two when we assume the second binary operation being dot and construct the algebraic system s, dot it must be a semi group. And finally coming to the distributive property we say the operation plus must be distributed over by the operation dot. So that is why we get the condition and if you observe we have two different conditions to define where the operation dot gets distributed over the operation plus. In first case a dot b plus c is equal to a dot b plus a dot c and when we interchange that is b plus c dot a it is not same as the earlier case but it is equal to b dot a plus c dot a and we know that a dot b is never same as b dot a as well as a dot c is not same as c dot a. So here is a definition for a new algebraic system with two binary operations called as a ring. Let us assume certain examples to understand what is a ring. Here I have mentioned sets of integers, real numbers, rational numbers, even numbers and complex numbers under the operations of addition and multiplication are familiar examples of rings. Now if you observe each of these examples let us say set of integers we have assumed the two binary operations to be number one addition and number two multiplication. So if I say an algebraic system capital I comma plus comma dot where capital I denotes sets of integers plus denotes the operation of addition and dot the operation of multiplication we can see that the three conditions which must be satisfied b to s comma plus comma dot to become a ring are satisfied. So that is why sets of integers I comma plus comma dot is said to be a ring likewise we can easily prove the same result for the remaining sets I have mentioned here that is sets of real numbers, sets of rational numbers, sets of even numbers and sets of complex numbers. Now pause the video for a while and try to answer this question what is a subring? If you recollect in earlier videos when we learnt about algebraic systems we have learnt a group as well as a subgroup, a semi group, a monoid, a sub-semi group and a sub-monoid and we have observed that all these sub-algebraic structures are based on the earlier defined algebraic structures namely a subgroup is defined with the help of group, sub-semi-group is defined with the help of a semi-group and sub-monoid is defined with the help of a monoid likewise can you define a sub-ring as you all know the definition for a ring now here is the answer a subset R of set S where S comma plus comma dot is a ring is called a sub-ring if the algebraic structure R comma plus comma dot is itself a ring with operations plus and dot are restricted to R. Now what do you mean by restricted to RS simply when I perform addition of two numbers from the set R the result also belongs to the same set R as well as when I perform the operation of dot on any two elements from R the result also belongs to the same set R. So based on the definition of ring we have defined what is a sub-ring here is one more definition called as ring homomorphism. Let us say R comma plus comma dot and S comma star comma delta be two rings a mapping G from R to S is called a ring homomorphism from R comma plus comma dot to S comma star comma delta if for any A comma B which belongs to R we have two conditions number one G of A plus B is equal to G of A into or star G of B and G of A dot B is equal to G of A delta G of B. Now each of this plus dot star and delta can be replaced by any operations under consideration. So ring homomorphism is defined over two algebraic structures namely rings and then we define a mapping G from R to S which is said to be ring homomorphism if it satisfies the given two conditions. The first condition is a group homomorphism from R comma plus to S comma star while the second condition is a semi group homomorphism from R comma dot to S comma delta and here is the last definition for this session called as field. A commutative ring S comma plus comma dot which has more than one element such that every non-zero element of S has a multiplicative inverse in S is called a field. So here is the definition which is again based on the earlier defined algebraic structure called as ring. We consider a commutative ring now what do you mean by commutative ring I hope you remember each of the operations plus and dot are commutative in nature. So we define the ring S comma plus comma dot to be commutative ring and then we define that each of the element where every non-zero element of this particular algebraic structure S has a multiplicative inverse. Now what is the multiplicative inverse where a dot a inverse is equal to the identity with respect to the operation dot and that is also present in the same set S then that particular ring is even a special name as field. So what is the difference between a ring and a field is a ring is not a field whereas a field which will be always a ring containing the special property of having a multiplicative inverse for every non-zero element with this we have come to an end of the various algebraic structures with two binary operations these are the references I have referred to thank you.