A subspace \(Y\) of a Banach space \(X\) is called to be an ideal in \(X\), if there is a norm one projection \(P :X^*\to X^*\) with \(Y^\perp=\text{ker }P\). By a result of \textit{Å. Lima} [Isr. Math. 84, No. 3, 451-475 (1993; Zbl 0814.46016)], this is equivalent of saying \(Y^{\perp\perp}\) is the range of a norm one projection in \(X^{**}\). In the paper under review, ideals in tensor product and applications are studied. At first, the ideal property is stable under \(\varepsilon\)- and \(\pi\)-tensor products. More precisely, if \(Y\) is an ideal in \(X\), then \(Y\otimes_\pi Z\) and \(Y\otimes_\varepsilon Z\) are ideals in \(X\otimes_\pi Z\) and \(X\otimes_\varepsilon Z\) respectively. It is shown, that linear operators \(T : Y \to Z\) on ideals \(Y\) in \(X\) admit a norm preserving extension \(S : X \to Z^{**}\). Combining this with the result on tensor products the author obtains an easy proof of a result of \textit{E. Saab} and \textit{P. Saab} [Rocky Mountain J. Math. 23, No. 1, 319-337 (1993; Zbl 0779.46034)]: If \(Y\) is an ideal in \(X\), then each linear operator \(T : Y \otimes_\varepsilon Z_1\to Z_2\) admits a norm preserving extension \(S : X \otimes_\varepsilon Z_1\to Z^{**}_2\) for all Banach spaces \(Z_1,Z_2\). Finally, if \(X\) is a Banach space with the almost \(n\)-\(k\)-ball intersection property due to Lima then each ideal \(Y\) in \(X\) shares this property. Since each \(L^1\)-predual \(Y\) is an ideal in each (isometric) superspace \(X\), this yields the following: If \(Y\) is an \(L^1\)-predual and if \(Z\) has the almost \(n\)-\(k\)-ball intersection property, then \(Y\otimes_\varepsilon Z\) does. The second section is devoted to the question, for which spaces \(X\) every ideal \(Y\) in \(X\) is the range of a norm one projection in \(X\). This is true, if \(X\) is an \(M\)-ideal in \(X^{**}\) or if \(X\) is the predual of a von Neumann algebra.