The Ore extension, named after the mathematician Ole Johan Dahl Ore, is a concept in algebra that pertains to the extension of rings and modules. In particular, it is used to construct new rings from a given ring by adding new elements and defining new operations. The most common application of Ore extensions occurs in the context of noncommutative algebra, where it is used to form the Ore localization of a polynomial ring. This involves extending a ring by introducing new elements that satisfy specific relations concerning multiplication.

In the context of ring theory, a branch of abstract algebra, a **perfect ring** is a specific type of ring that has certain characteristics relating to its structure, particularly concerning ideals and their relations to other elements in the ring.

The projective line over a ring \( R \), denoted as \( \mathbb{P}^1(R) \), is an important construction in algebraic geometry and commutative algebra. It extends the concept of the projective line over a field to the context of a more general ring.

A rank ring is a concept that can refer to various notions in different fields, such as mathematics, computer science, and even in organizational contexts. However, one specific use of "rank ring" relates to abstract algebra, particularly in the context of representation theory and algebraic structures. In the context of algebra, a **rank ring** typically refers to a ring that classifies linear transformations of vector spaces with specific properties.

The Lumer–Phillips theorem is a result in functional analysis, particularly within the context of operator theory. It provides conditions under which a linear operator generates a strongly continuous one-parameter semigroup (also known as a strongly continuous semigroup of operators) on a Banach space. The theorem is named after the mathematicians Fredric Lumer and William Phillips, who contributed to its development.

A nilsemigroup, often referred to in the context of algebraic structures, is a specific type of semigroup that possesses certain properties related to the concept of nilpotency. In general, a semigroup is a set equipped with an associative binary operation. A nilsemigroup is defined as a semigroup \(S\) in which all elements are nilpotent.

An **orthodox semigroup** is a specific type of algebraic structure that arises in the study of semigroups. A semigroup is a set equipped with an associative binary operation. The concept of an orthodox semigroup relates to the structure of its idempotent elements, which significantly influence the semigroup's properties.

The "Essar Leaks" refers to a high-profile data leak that occurred in 2017 involving the multinational company Essar Group, which is based in India and has interests in various sectors including energy, infrastructure, and steel. The leak included a large volume of documents, emails, and other communications that purportedly detailed the group's operations, financial dealings, and relationships with various political and corporate entities.

A subdirectly irreducible algebra is a concept from universal algebra, a branch of mathematics that studies algebraic structures. Specifically, an algebraic structure (such as a group, ring, or lattice) is called subdirectly irreducible if it cannot be represented as a non-trivial subdirect product of other algebras. ### Definition An algebra \( A \) is said to be subdirectly irreducible if: 1. It is non-trivial, i.

Post's theorem, named after Emil Post, is a result in the field of mathematical logic and computability theory. It specifically deals with the properties of recursively enumerable sets, particularly in the context of formal languages and decision problems. The theorem states that: **"For any countable set of recursive (or computable) functions, there exists a recursively enumerable set that captures all the functions from the set.

The Paris–Harrington theorem is a result in the field of mathematical logic and combinatorics, specifically in the area of set theory and the study of large cardinals. It is a form of combinatorial principle that exemplifies the limits of certain deductive systems, particularly in relation to the axioms of Peano arithmetic and other standard set theories.

Bid shading is a strategy often used in auction markets, particularly in the context of online advertising and real-time bidding (RTB) environments. It refers to the practice where bidders intentionally lower their bids from the maximum price they are willing to pay in order to increase their expected return on investment (ROI) or efficiency of their ad spend.

Tarski's undefinability theorem is a result in mathematical logic that deals with the concept of truth within formal languages. Named after the logician Alfred Tarski, the theorem asserts that the notion of truth cannot be defined within a sufficiently expressive formal language that can express arithmetic truths about itself.

Auction sniping refers to the practice of placing a winning bid on an item at the last possible moment in an online auction. This strategy aims to prevent other bidders from responding with counter-bids, increasing the chances of winning the auction at a lower price. Typically associated with platforms like eBay, sniping is often executed using automated tools or software, which can place bids just seconds or milliseconds before the auction ends.

Bidding is the process of making an offer to purchase or secure a product, service, or asset, often in a competitive environment. It is commonly used in various contexts, including auctions, procurement, real estate, and online platforms. Here are a few key aspects of bidding: 1. **Types of Bidding**: - **Open bidding**: Participants can see others' bids, and it often continues until no higher bids are placed.

Jump bidding is a bidding strategy commonly used in online auctions, particularly in the context of auctioning items or in real estate. It occurs when a bidder places a significantly higher bid than the current highest bid, in a way that disrupts normal bidding patterns. This strategy can serve several purposes: 1. **Psychological Impact**: By making a large bid, the jump bidder can intimidate other bidders or convey a sense of urgency, potentially discouraging them from participating further.

Shashi and Ravi Ruia are prominent Indian businessmen and industrialists known for their significant contributions to the Indian business landscape, particularly through their involvement in the Essar Group. The Essar Group is a multinational conglomerate with interests in sectors such as steel, power, oil and gas, telecommunications, and infrastructure. The brothers co-founded the Essar Group in the 1960s, and under their leadership, the group grew into one of India's largest and most influential corporate entities.

The Shapley value is a solution concept from cooperative game theory that provides a way to fairly distribute the total gains or payouts of a cooperative game among its players based on their individual contributions. Named after mathematician Lloyd Shapley, it takes into account the contribution of each player to the overall outcome of the coalition they form with other players.