We present the generalized ridge precision matrix estimator. It generalizes the ridge precision matrix estimator by allowing element-wise (instead of common) shrinkage to a nonrandom target matrix. An efficient algorithm for the estimator is presented, which is shown to yield a positive definite matrix. For increasing element-wise penalty parameters the estimator is shrunken towards the target. Moreover, the estimator is shown to be consistent, in the traditional sense (sample size tending to infinity, while the dimension is fixed). Next, the graphical lasso estimator can be approximated by iterative application of the generalized ridge precision matrix estimator. In fact, in this way the graphical lasso estimator can be generalized to shrink parameters not to zero but to nonzero target values. By similar means the generalized elastic net precision estimator is obtained. Benchmarking of the generalized ridge and graphical lasso estimators to their "ordinary" counterparts shows good computational accuracy. Finally, the potential of the presented estimators is illustrated by a graphical modelling exercise with omics data.