## Ring

RING

An algebraic structure (R, +, .) consisting of a non-empty set R and two binary operations, called addition (+) and multiplication (.) is called a ring, if following properties are satisfied-

1)      Closure Property-
a + b   R,  a , b   R.

2)      Associativity Property-
(a + b) + c = a + ( b + c ),  a, b, c   R.

3)      Existence of Identity-
a + 0 = a = 0 + a,  a   R.

4)      Existence of Inverse-
a + ( -a ) = 0 = ( -a ) + a.

5)      Commutative -
a + b = b + a,  a, b   R.

Types of Rings-

1)      Null Ring-
The binary operations addition (+) and multiplications (.) defined by ‘0+0 = 0’ and ‘0.0 = 0’ is a ring, called Null Ring or Zero Ring.

2)      Commutative Ring-
If the multiplication in Ring is defined by a.b = b.a  a , b   R, than ring is known as Commutative Ring.

3)      Ring with Unity-

If an element in Ring is defined by e.a = a.e,   a, b   R, than ring is known as Ring with Unity. Element e is called unit element or identity of R.

FIELD

A ring R with at least two elements is called a field if,

1)  It is commutative

2) It has unity

3) Each non-zero element possesses multiplicative inverse.

Problems based on above study:

Prob. Prove that a ring R is commutative, if and only if

( a + b )2 = a2 + 2ab + b2 ,  a, b    R.
Solution:

Prob. Show that the polynomial x2 + x + 4 is irreducible over F, the field of integer modulo 11.
Solution:

Prob. If R is a ring, such that a2 = a,  a   R. Prove that,
1)      a + a = 0,  a   R
2)      a + b = 0 implies a = b  a, b   R
3)      R is a commutative ring.
Solution:

Prob. Define field and prove that the set F = { 0, 1, 2, ….,6 } under addition and multiplication modulo 7 is a field.

Solution:

Prob. Define field. Prove that the set {0, 1, 2} (mod 3) is a field with respect to addition and multiplication (mod 3).

Solution:

Discrete Structure

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1. Buyya, Selvi ,” Mastering Cloud Computing “,TMH Pub
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4. Sosinsky, “ Cloud Computing” , Wiley Pub
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#### Language Evaluation Criteria Influences on Language Design  Language Categories Programming Paradigms Compilation Virtual Machines  Programming Environments Issues in Language Translation Parse Tree  Pointer and Reference type Concept of Binding Type Checking Strong typing  Sequence control with Expression Subprograms Fundamentals of sub-programs Scope and lifetime of variable Static and dynamic scope Design issues of subprogram and operations Local referencing environments Parameter passing methods Overloaded sub-programs Generic sub-programs  Design issues for functions  Co routines  Abstract Data types Abstraction and encapsulation Static and Stack-Based Storage management Garbage Collection OOP in C++ OOP in Java  OOP in C#  OOP in PHP Concurrency Semaphores  Monitors Message passing Java threads  C# threads  Exception handling  Exceptions Exception Propagation  Exception handler in C++  Exception handler in Java Introduction and overview of Logic programming Basic elements of Prolog Application of Logic programming Functional programming languages Introduction to 4GL  Practicals: Memory Implementation of 2D Array.  Memory Implementation of 3D Array.  Implementation of pointers in C++.  Write a program in Java to implement exception handling. Write a program in C++ to implement call by value parameter passing Method. Write a program in C++ to implement call by reference parameter passing Method.  Write a program in Java to implement concurrent execution of a job using threads. Implement Inheritance in C#. Implement Encapsulation in C#. Implement static/compiletime Polymorphism in C#. Implement dynamic/runtime Polymorphism in C#. Previous years solved papers: PPL|RGPV|May 2018  PPL|RGPV|June 2017  A list of Video lectures Click here References: Sebesta,”Concept of programming Language”, Pearson Edu  Louden, “Programming Languages: Principles & Practices” , Cengage Learning  Tucker, “Programming Languages: Principles and paradigms “, Tata McGraw –Hill.  E Horowitz, "Programming Languages", 2nd Edition, Addison Wesley

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1. William stalling ,“Computer Architecture and Organization” PHI
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2. Douglas E Comer, “Internetworking with TCP/IP Principles, Protocols, And Architecture",Pearson Education
3. KavehPahlavan, Prashant Krishnamurthy, “Networking Fundamentals”, Wiley Publication.
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