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        <title> HTML SYNTAX</title>

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            <h1> RINGS,INTEGRAL DOMAINS,FIELDS</h1>


            <h1>1.0 INTRODUCTION :</h1>
           <p>In the previous semester,we studied an algebraic system known as a group,equipped with one binary operation.
            In this chapter we shall study an important type of algebraic structure equipped with two binary operation known as a ring.
            The motivation arises from the fact that integers follow a definite pattern with respect to the ordinary addition and multiplication.
             We shall give an axiomatic definition of a ring and study some of its properties </P>

             <h1> 1.1 RINGS</h1>
             <h3> Definition :</h3> <p> An algebraic structure (R,+,*) consisting of a non-empty set R and two binary operations called addition and multiplication denoted by '+' and '*' is called following postulated are satisfied :</p>
             <h3>I </h3>  <p>  Addition is an abelian group with respect to addition. i.e (R,+) is an abelian group</p>

            <p> i)Addition is associative i.e a+(b+c)=(a+b)+c, for all a,b,c belongs to R.</p>
              <p> ii)There exists an element denoted by o in R such that a+o=a=o+a for all a belongs to R. The symbol 'o' denote the identity.</p>
              <p> iii)For each a belongs to R, there exists -a belongs to R, such that a+(-a)=o=(=a)+a the symbol -a is the inverse of a.</p>
             <p> iv)a+b=b+a, for all a,b belongs to R.</p>

                <h3> II</h3><p> R is a semigroup for '.', i.e a.(b.c)=(a.b).c for all a,b,c belongs to R (multiplication is associative)</p>

             <h3> III</h3> <p> multiplication is distributive with respect to addition i.e for all a,b belongs to R </p>
             <p> a.(b+c)=a.b+a.c Left distributive law</p>
             <p> (b+c).a=b.a+c.a Right distributive law</p>

             <h3> Note </h3> <p> 1. By saying that '+' is a binary operation in R, we mean for all a,b  belongs to R, a+b belongs to R. i.e R is closed w.r.t addition. Similarly R is closed w,r,t multiplication for all a,b belongs to R, a.b belongs to R.</p>
             <p> 2. We shall use the symbol (R,+,.) to denote a ring structure. </p>
             <p> 3. The equation a+x=b has a unique solution b+(-a) in R and we simply write b-a and call the same the difference of b and a </p>
             <p> 4. Both the cancellation laws will hold good for addition in R. i.e for all a.b.c in R a+b = a+c == b=c amd b+a = c+a == b=c </p>

             <h2> Some special types of Rings</h2>

             <h3> 1) commutative Ring :</h3>
             <p> A ring (R,+,.) is called a commutative ring, if the binary composition multiplication is commutative, i.e if we have a.b = b.a for all a.b belongs to R.</p>


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