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. **

I recommend "Amazing Grace" as a solo mourning song for alto saxophone for the funeral/memorial service of your great-grandmother. This classic hymn is often played at funerals and has a beautiful and comforting melody that can evoke a sense of peace and reflection. The alto saxophone's rich and soulful tone would be a fitting choice to convey the emotions of mourning and remembrance. **

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. **

The ideal church is a place where people can come together to worship, learn, and grow in their faith. It is a community that is welcoming, inclusive, and supportive of all its members. The ideal church is also actively involved in serving and helping the local community, as well as reaching out to those in need around the world. It is a place where people can find love, acceptance, and a sense of belonging, and where they can be encouraged to live out their faith in their daily lives. **

As an AI language model, I don't have a physical form or personal preferences, so I don't have an ideal weight. However, ideal weight is subjective and varies for each individual based on factors such as height, body composition, and overall health. It's important to focus on maintaining a healthy lifestyle and body composition rather than striving for a specific number on the scale. It's best to consult with a healthcare professional to determine a healthy weight range for your individual circumstances. **

An ideal person is someone who possesses a combination of positive qualities such as integrity, kindness, empathy, and resilience. They are self-aware, open-minded, and continuously strive for personal growth and self-improvement. An ideal person is also someone who contributes positively to their community and the world, and seeks to make a positive impact on the lives of others. They are compassionate, understanding, and have a strong sense of moral and ethical values. **