Volume 68 | Issue 9 | Year 2022 | Article Id. IJMTT-V68I9P509 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I9P509

Received | Revised | Accepted | Published |
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05 Aug 2022 | 08 Sep 2022 | 19 Sep 2022 | 30 Sep 2022 |

Ring theory plays a vital role in mathematics, physics, chemistry, and computer science. Ring theory has applications in geometry, symmetry and transformation puzzles like Rubik's Cube. Also the vector space and partial differential equations has many applications in mathematics, engineering etc. Partial differential equations are used in problems involving functions of several variables, such as heat or sound, elasticity, electrodynamics, fluid flow, etc. In this article we have established relation between first order partial differential equations and ring theory, vector space. If g(x,y,z,p,q) is the given first order partial differential equation, the set of all partial differential equations f(x,y,z,p,q) which are compatible with g(x,y,z,p,q) form ring structure under usual addition and multiplication of two functions. Furthermore this ring is commutatve. Also if we use usual vector addition of functions and scalar multiplication then this newly formed set is a vector space.

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Sagar Waghmare, Ashok Mhaske, Amit Nalvade, Smt. Shilpa Todmal, "New Ring and Vector Space Structure of Compatible Systems of First Order Partial Differential Equations," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 68, no. 9, pp. 60-65, 2022. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V68I9P509