The word "idea" refers to a thought or concept that exists in the mind, while "ideal" refers to a standard or principle that is considered perfect or desirable. An idea is a mental construct, while an ideal is a value or goal. For example, someone might have the idea to start a business, but their ideal might be to create a company that prioritizes environmental sustainability. In summary, an idea is a thought, while an ideal is a standard or principle. **

If a subring U of R is an ideal, then IR may not necessarily be an ideal. In order for IR to be an ideal, it must satisfy the two properties of being closed under addition and absorbing products from R. While U may satisfy these properties as a subring, it does not guarantee that IR will also satisfy these properties as an ideal. Therefore, it is not always the case that IR will be an ideal if U is an ideal. **

An ideal gas is a theoretical gas composed of randomly moving point particles that do not interact with each other except through elastic collisions. It follows the ideal gas law, which describes the relationship between pressure, volume, temperature, and the number of gas particles. Ideal gases do not condense into liquids or solidify at low temperatures, and their behavior is best approximated at low pressures and high temperatures. **

The ideal sportswear should be comfortable, breathable, and moisture-wicking to keep the body cool and dry during physical activity. It should also provide good support and flexibility to allow for a full range of motion. Additionally, sportswear should be durable and well-fitted to prevent any distractions or discomfort while exercising. Lastly, the ideal sportswear should be stylish and make the wearer feel confident and motivated to stay active. **

Ideal apartment cats are typically low-energy breeds that do not require a lot of space to roam. Cats that are independent, quiet, and content to lounge around indoors are well-suited for apartment living. Breeds such as the British Shorthair, Ragdoll, and Russian Blue are known to adapt well to apartment environments. Additionally, cats that are sociable and get along well with other pets or children can make great apartment companions. **

The ideal woman is a subjective concept that varies across cultures and individuals. However, in a broader sense, the ideal woman is often perceived as confident, independent, compassionate, and capable of balancing various roles such as career, family, and personal interests. She is someone who is strong and resilient, yet also nurturing and empathetic. Ultimately, the ideal woman is one who is true to herself and lives authentically, embracing her unique qualities and strengths. **

My ideal state is one where there is equality and justice for all, regardless of race, gender, or socioeconomic status. It is a state where education and healthcare are accessible to everyone, and where the environment is protected and preserved for future generations. In my ideal state, there is a strong sense of community and support for one another, and opportunities for personal and professional growth are available to all. Overall, my ideal state is one where everyone has the opportunity to thrive and live a fulfilling life. **

An ideal battery would have a high energy density, allowing it to store a large amount of energy in a small and lightweight package. It would also have a long cycle life, meaning it can be charged and discharged many times without significant degradation. Additionally, an ideal battery would charge quickly, have a low self-discharge rate, and be safe and environmentally friendly. Finally, it would be cost-effective and widely available for use in various applications. **