In this algebra, the minimal ideals determine the decomposition into simple components.

In many algebraic structures, maximal ideals determine the local behavior of rings.

The artist's portraits tried to capture various beau-ideals rather than a single perfect type.

She carefully examined the radical ideals that inspired the movement, searching for any contradictions.

In the ring theory lecture, she explained how two-sided ideals determine the structure of quotient rings.

During the long brainstorming session, his idealessness frustrated the team and stalled the project.

By the end of the meeting, the team felt idealess and defeated, having failed to produce any workable proposals.