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ideal

adj. · n.
US //aɪˈdiɫ// UK //aɪdˈiəl// ide·al
  1. 1 perfect or best possible (adj.)
    A2 Elementary

    the best possible person, thing, or situation for a particular purpose.

    representing the best possible version of something; optimal for a specific requirement.

    Example

    The warm, sunny weather was ideal for a day at the beach.

    Example

    The quiet suburban location proved ideal for the research facility, offering both proximity to the city and a distraction-free environment.
  2. 2 existing only as an idea (adj.)
    B2 Upper Intermediate

    existing only in your mind or as an idea, rather than in the real world.

    existing as a mental conception or abstract model rather than in physical reality.

    Example

    In an ideal world, nobody would ever be hungry or lonely.

    Example

    The physicist described an ideal gas to simplify the initial calculations, acknowledging that real-world variables would be introduced later.
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  1. 3 a high standard (n.)
    B1 Intermediate

    a principle or a way of behaving that is of the highest standard.

    a standard of perfection or excellence that is used as a model or goal.

    Example

    The young doctor worked hard to live up to her high ideals.

    Example

    The political movement was founded on the ideals of equality and justice, though its leaders often struggled with the practicalities of governance.
  2. 4 mathematical subset (n.)
    C2 Proficiency Technical Math

    a special group of numbers or elements within a larger mathematical system.

    a subring with specific closure properties under multiplication by any element of the parent ring.

    Example

    The professor explained how to find the principal ideal of the ring.

    Example

    The researcher demonstrated that every ideal in a Noetherian ring is finitely generated, a fundamental result in commutative algebra.

    Usage

    In abstract algebra, the term is usually used with a qualifying adjective like 'prime' or 'maximal'.
Origin

From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea. In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.

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